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SPLENDID MORITA EQUIVALENCES FOR PRINCIPAL 2-BLOCKS

WITH DIHEDRAL DEFECT GROUPS

SHIGEO KOSHITANI AND CAROLINE LASSUEUR

Abstract Given a dihedral 2-group P of order at least 8 we classify the splendidMorita equivalence classes of principal 2-blocks with defect groups isomorphic to P Tothis end we construct explicit stable equivalences of Morita type induced by specificScott modules using Brauer indecomposability and gluing methods we then determinewhen these stable equivalences are actually Morita equivalences and hence automati-cally splendid Morita equivalences Finally we compute the generalised decompositionnumbers in each case

1 Introduction

In this paper we are concerned with the classification of principal 2-blocks with dihedraldefect groups of order at least 8 up to splendid Morita equivalence also often called Puigequivalence

This is motivated by a conjecture of Puigrsquos [Pui82] known as Puigrsquos Finiteness Con-jecture (see Broue [Bro94 62] or Thevenaz [The95 (386) Conjecture] for publishedversions) stating that for a given prime p and a finite p-group P there are only finitelymany isomorphism classes of interior P -algebras arising as source algebras of p-blocks offinite groups with defect groups isomorphic to P or equivalently that there are only afinite number of splendid Morita equivalence classes of blocks of finite groups with defectgroups isomorphic to P This obviously strengthens Donovanrsquos Conjecture However weemphasise that by contrast to Donovanrsquos Conjecture if p is a prime number (KO k)a p-modular system with k algebraically closed and Puigrsquos Finiteness Conjecture holdsover k then it automatically holds over O since the bimodules inducing splendid Moritaequivalences are liftable from k to O

The cases where P is either cyclic [Lin96b] or a Klein-four group [CEKL11] are theonly cases where this conjecture has been proved to hold in full generality Else underadditional assumptions Puigrsquos Finiteness Conjecture has also been proved for severalclasses of finite groups as for instance for p-soluble groups [Pui94] for symmetric groups[Pui94] for alternating groups [Kes02] of the double covers thereof [Kes96] for Weylgroups [Kes00] or for classical groups [HK00 HK05 Kes01] The next cases to investigateshould naturally be in tame representation type

Date March 8 20192010 Mathematics Subject Classification 16D90 20C20 20C15 20C33Key words and phrases Puigrsquos finiteness conjecture Morita equivalence splendid Morita equivalence

stable equivalence of Morita type Scott module Brauer indecomposability generalised decompositionnumbers dihedral 2-group

The first author was supported by the Japan Society for Promotion of Science (JSPS) Grant-in-Aidfor Scientific Research (C)15K04776 2015ndash2018 The second author acknowledges financial support bythe TU Nachwuchsring of the TU Kaiserslautern as well as by DFG SFB TRR 195

1

2 S Koshitani and C Lassueur

In this paper we investigate the principal blocks of groups G with a Sylow 2-subgroup Pwhich is dihedral of order at least 8 over an algebraically closed field k of characteristic 2Our main result is the following

Theorem 11 Let n ge 3 be a positive integer Then the splendid Morita equivalenceclasses of principal 2-blocks of finite groups with dihedral defect group D2n of order 2n

coincide with the Morita equivalence classes of such blocks More accurately a principalblock with dihedral defect group D2n is splendidly Morita equivalent to precisely one of thefollowing blocks

(1) kD2n(2) B0(kA7) in case n = 3(3) B0(k[PSL2(q)]) where q is a fixed odd prime power such that (q minus 1)2 = 2n(4) B0(k[PSL2(q)]) where q is a fixed odd prime power such that (q + 1)2 = 2n(5) B0(k[PGL2(q)]) where q is a fixed odd prime power such that 2(q minus 1)2 = 2n or(6) B0(k[PGL2(q)]) where q is a fixed odd prime power such that 2(q + 1)2 = 2n

In particular if q and qprime are two odd prime powers as in (3)-(6) such that either(q minus 1)2 = (qprime minus 1)2 (Cases (3) and (5)) or (q + 1)2 = (qprime + 1)2 (Cases (4) and (6)) thenB0(k[PSL2(q)]) is splendidly Morita equivalent to B0(k[PSL2(q

prime)]) and B0(k[PGL2(q)]) issplendidly Morita equivalent to B0(k[PGL2(q

prime)])

Remark 12 We note that if G is a soluble group and B is an arbitrary 2-block of Gwith a defect group P sim= D2n with n ge 3 which is not nilpotent then n = 3 and B isactually splendidly Morita equivalent to kS4

sim= k[PGL2(3)] (see [Kos82]) There is alsoan interesting and related result by Linckelmann [Lin94] where all derived equivalenceclasses of blocks B with dihedral defect groups over the field k are determined

Furthermore we will prove in Corollary 47 that for a given defect group P sim= D2n

(n ge 3) up to stable equivalence of Morita type there are exactly three equivalenceclasses of principal blocks of finite groups G with defect group P and these depend onlyon the fusion system FP (G) or equivalently on the number of modular simple modulesin B0(kG)

In order to prove Theorem 11 we will construct explicit Morita equivalences inducedby bimodules given by Scott modules of the form Sc(GtimesGprime∆P ) First we will constructstable equivalences of Morita type using these modules using gluing methods and thendetermine when these stable equivalences are actually Morita equivalences To reach thisaim we make use of the notion of Brauer indecomposability introduced in [KKM11] Inparticular we will use some recent results of Ishioka and Kunugi [IK17] in order to provethe following theorem

Theorem 13 Let G be a finite group with a dihedral 2-subgroup P of order at least 8Assume moreover that the fusion system FP (G) is saturated and CG(Q) is 2-nilpotentfor every FP (G)-fully normalised non-trivial subgroup Q of P Then the Scott moduleSc(G P ) is Brauer indecomposable

This result crucial for our work may in fact be of independent interest as it is anextension of the main results of [KKL15] We note that further results on Brauer in-decomposability of Scott modules under different hypotheses may be found in [KKM11Theorem 12] [KKL15 Theorem 12(b)] and [Tuv14]

Splendid Morita equivalences for principal 2-blocks with dihedral defect groups 3

The paper is structured as follows In Section 2 we set up our notation and recall back-ground material which we will use throughout In Section 3 we establish some propertiesof Scott modules of direct products with respect to diagonal p-subgroups From Section 4onwards we will assume that the field k has characteristic 2 and we will focus our atten-tion on groups with dihedral Sylow 2-subgroups of order at least 8 In Section 4 we proveTheorem 13 In Sections 5 and 6 we determine when the stable equivalences constructedin Section 4 are indeed Morita equivalences In Section 7 we prove Theorem 11 andfinally in Section 8 as a consequence of Theorem 11 we can specify the signs occurringin Brauerrsquos computation of the generalised decomposition numbers of principal blockswith dihedral defect groups in [Bra66 sectVII] This will yield the following result

Corollary 14 If G is a finite group with a dihedral Sylow 2-subgroup of order 2n withn ge 3 then |Irr(B0(kG))| = 2nminus2 + 3 and the values at non-trivial 2-elements of theordinary irreducible characters in Irr(B0(kG)) are given by the non-trivial generaliseddecomposition numbers of B0(kG) and depend only on the splendid Morita equivalenceclass of B0(kG)

Here by non-trivial generalised decomposition number we mean the generalised de-composition numbers parametrised by non-trivial 2-elements

2 Notation and quoted results

21 Notation Throughout this paper unless otherwise stated we adopt the followingnotation and conventions All groups considered are assumed to be finite and all modulesover finite group algebras are assumed to be finitely generated unitary right modules Welet G denote a finite group and k an algebraically closed field of characteristic p gt 0

Given a positive integer n we write D2n CnSn and An for the dihedral group oforder 2n the cyclic group of order n the symmetric group of degree n and the alternatinggroup of degree n respectively We write H le G when H is a subgroup of G Giventwo finite groups N and H we denote by N ⋊ H a semi-direct product of N by H(where N ⊳ (N ⋊H)) For a subset S of G we set Sg = gminus1Sg and for h isin G we sethg = gminus1hg For an integer n ge 1 〈g1 gn〉 is the subgroup of G generated by theelements g1 gn isin G We denote the centre of G by Z(G) and we set ∆G = (g g) isinGtimesG | g isin G le GtimesG

Given a p-subgroup P le G we denote by FP (G) the fusion system of G on P that isthe category whose objects are the p-subgroups of P and whose morphisms from Q to Rare the group homomorphisms induced by conjugation by elements of G see [AKO11Definition I21] We recall that if P is a Sylow p-subgroup of G then FP (G) is saturatedsee [AKO11 Definition I22] For further notation and terminology on fusion systemswe refer to [AKO11] and [BLO03]

The trivial kG-module is denoted by kG If H le G is a subgroup M is a kG-moduleand N is a kH-module then we write Mlowast = Homk(M k) for the k-dual of M MdarrHfor the restriction of M to H and NuarrG for the induction of N to G Given an H le Gwe denote by PH(kG) the H-projective cover of the trivial module kG and we let ΩH(kG)denote the H-relative Heller operator that is ΩH(kG) = Ker (PH(kG) kG) the kernelof the canonical projection (see [The85]) We write B0(kG) for the principal block of kGFor a p-block B we denote by Irr(B) the set of ordinary irreducible characters in B andby IBr(B) the set of irreducible Brauer characters in B Further we use the standardnotation k(B) = |Irr(B)| and l(B) = |IBr(B)|


For a subgroup H le G we denote the (Alperin-)Scott kG-module with respect to Hby Sc(GH) By definition Sc(GH) is the unique indecomposable direct summand ofthe induced module kHuarr

G which contains kG in its top (or equivalently in its socle) IfQ isin Sylp(H) then Q is a vertex of Sc(GH) and a p-subgroup of G is a vertex of Sc(GH)if and only if it is G-conjugate to Q It follows that Sc(GH) = Sc(GQ) We refer thereader to [Bro85 sect2] and [NT88 Chap4 sect84] for these results Furthermore we willneed the fact that Sc(GH) is nothing else but the relative H-projective cover PH(kG) ofthe trivial module kG see [The85 Proposition 31] In order to produce splendid Moritaequivalences between principal blocks of two finite groups G and Gprime with a common defectgroup P we mainly use Scott modules of the form Sc(GtimesGprime ∆P ) which are obviously(B0(kG) B0(kG

prime))-bimodules by the previous remarkFor further notation and terminology we refer the reader to the books [Gor68] [NT88]

and [The95]

22 Equivalences of block algebras Let G and H be two finite groups and let Aand B be block algebras of kG and kH with defect groups P and Q respectively

The algebras A and B are called splendidly Morita equivalent (or Puig equivalent)if there is a Morita equivalence between A and B induced by an (AB)-bimodule Msuch that M seen as a right k[G times H ]-module is a p-permutation module In thiscase we write A simSM B Due to a result of Puig (see [Pui99 Corollary 74] and [Lin18Proposition 971]) the defect groups P and Q are isomorphic (and hence from now on weidentify P and Q) Obviously M is indecomposable as a k(GtimesH)-module Further since

AM andMB are both projective M has a vertex R which is written as R = ∆(P ) le GtimesH Then this is equivalent to the condition that A and B have source algebras which areisomorphic as interior P -algebras by the result of Puig and Scott (see [Lin01 Theorem 41]and [Pui99 Remark 75])

In particular we note that if the Scott module M = Sc(GtimesH∆P ) induces a Moritaequivalence between the principal blocks A and B of kG and kH respectively then thisis a splendid Morita equivalence because Scott modules are p-permutation modules bydefinition

Let now M be an (AB)-bimodule and N a (BA)-bimodule Following Broue [Bro94sect5] we say that the pair (MN) induces a stable equivalence of Morita type between Aand B if M and N are projective both as left and right modules and there are isomor-phisms of (AA)-bimodules and (BB)-bimodules

M otimesB N sim= AoplusX and N otimesA M sim= B oplus Y

respectively where X is a projective (AA)-bimodule and Y is a projective (BB)-bimodule

The following result of Linckelmann will allow us to construct Morita equivalences usingstable equivalences of Morita type

Theorem 21 ([Lin96 Theorem 21(ii)(iii)]) Let A and B be finite-dimensional k-algebras which are indecomposable non-simple self-injective k-algebras Let M be an(AB)-bimodule inducing a stable equivalence between A and B

(a) If M is indecomposable then for any simple A-module S the B-module S otimesA Mis indecomposable and non-projective

(b) If for any simple A-module S the B-module S otimesA M is simple then the functorminusotimesA M induces a Morita equivalence between A and B


Furthermore we recall the following fundamental result originally due to Alperin[Alp76] and Dade [Dad77] which will provide us with an important source of splendidMorita equivalences in Section 6

Theorem 22 ([KK02 (31) Lemma]) Let G be a finite group Let G E G be a normal

subgroup such that GG is a pprime-group and G = GCG(P ) where P is a Sylow p-subgroup

of G Furthermore let e and e be the block idempotents corresponding to B0(kG) andB0(kG) respectively Then the following holds

(a) The map B0(kG) minusrarr B0(kG) a 7rarr ae is a k-algebra isomorphism so that ee =ee = e

(b) The right k[G times G]-module B0(kG) = ekG = ekGe induces a splendid Morita

equivalence between B0(kG) und B0(kG)

23 The Brauer construction and Brauer indecomposability Given a kG-module V and a p-subgroup Q le G the Brauer construction (or Brauer quotient) of Vwith respect to Q is defined to be the kNG(Q)-module

V (Q) = V Q sum

RltQ

TrQR(VR)

where V Q denotes the set of Q-fixed points of V and for each proper subgroup R lt QTrQR V R minusrarr V Q v 7rarr

sumxRisinQR xv denotes the relative trace map See eg [The95

sect27] We recall that the Brauer construction with respect to Q sends a p-permutationkG-module V functorially to the p-permutation kNG(Q)-module V (Q) see [Bro85 p402]

Furthermore following the terminology introduced in [KKM11] a kG-module V is said

to be Brauer indecomposable if the kCG(Q)-module V (Q) darrNG(Q)CG(Q) is indecomposable or

zero for each p-subgroup Q le GIn order to detect Brauer indecomposability we will use the following two recent results

of Ishioka and Kunugi

Theorem 23 ([IK17 Theorem 13]) Let G be a finite group and P a p-subgroup of G LetM = Sc(GP ) Assume that the fusion system FP (G) is saturated Then the followingassertions are equivalent

(i) M is Brauer indecomposable

(ii) Sc(NG(Q) NP (Q))darrNG(Q)QCG(Q) is indecomposable for each FP (G)-fully normalised sub-

group Q of P

Theorem 24 ([IK17 Theorem 14]) Let G be a finite group and P a p-subgroup ofG Q an FP (G)-fully normalised subgroup of P and suppose that FP (G) is saturatedAssume moreover that there exists a subgroup HQ of NG(Q) satisfying the following twoconditions

(1) NP (Q) isin Sylp(HQ) and(2) |NG(Q) HQ| = pa (a ge 0)

Then Sc(NG(Q) NP (Q))darrNG(Q)QCG(Q) is indecomposable


24 Principal blocks with dihedral defect groups Since O2prime(G) acts triviallyon the principal block of G it is well-known that B0(G) and B0(GO2prime(G)) are Moritaequivalent Such a Morita equivalence is induced by the (B0(k[GO2prime(G)]) B0(kG))-bimodule B0(k[GO2prime(G)]) which is obviously a 2-permutation module Hence theseblocks are indeed splendidly Morita equivalent and we may restrict our attention to thecase O2prime(G) = 1

We recall that if G is a finite group with a dihedral Sylow 2-subgroup P of order atleast 8 then Gorenstein and Walter [GW65] proved that GO2prime(G) is isomorphic to either

(D1) P (D2) the alternating group A7 or(D3) a subgroup of PΓL2(q) containing PSL2(q) where q is a power of an odd prime

In other words one of the following groups(i) PSL2(q)⋊Cf where q is a power of an odd prime such that q equiv plusmn1 (mod 8)

and f ge 1 is a suitable odd number or(ii) PGL2(q)⋊Cf where q is a power of an odd prime and f ge 1 is a suitable odd

number

The fact that q is a power of an odd prime can be found in [Suz86 Chapter 6 (89)]Moreover the splitting of case (D3) into (i) and (ii) follows from the fact that

PΓL2(q) sim= PGL2(q)⋊Gal(FqFr)

where q = rm is a power of an odd prime r and the Galois group Gal(FqFr) is cyclic oforder m generated by the Frobenius automorphism F Fq minusrarr Fq x 7rarr xr This impliesthat Cf is a cyclic subgroup of Gal(FqFr) generated by a power of F and moreover therequirement that P is dihedral forces f to be odd (Here we apply [Suz86 Chapter 6(89)] implying that Cf is of odd order See also the beginning of [Gor68 sect163])

25 Dihedral 2-groups and 2-fusion Assume G is a finite group having a Sylow2-subgroup P which is a dihedral 2-group D2n of order 2n (n ge 3) Write

P = D2n = 〈s t | s2nminus1

= t2 = 1 tst = sminus1〉

and set z = s2nminus2

so that 〈z〉 = Z(P ) Then there are three possible fusion systemsFP (G) on P

(1) First case FP (G) = FP (P ) There are exactly three G-conjugacy classes of in-volutions in P z s2jt | 0 le j le 2nminus2 minus 1 and s2j+1t | 0 le j le 2nminus2 minus 1Moreover l(B0(kG)) = 1 that is B0(kG) possesses exactly one simple modulenamely the trivial module kG

(2) Second case FP (G) = FP (PGL2(q)) where 2(q plusmn 1)2 = 2n There are exactlytwo G-conjugacy classes of involutions in P represented by the elements z and stNote that t is fused with z in this case Moreover l(B0(kG)) = 2

(3) Third case FP (G) = FP (PSL2(q)) where (q plusmn 1)2 = 2n There is exactly oneG-conjugacy class of involutions in P represented by z Moreover l(B0(kG)) = 3

In fact if P is a 2-subgroup of G but not necessarily a Sylow 2-subgroup and FP (G) issaturated then FP (G) is isomorphic to one of the fusion systems in (1) (2) and (3) Werefer the reader to [Gor68 sect77] [Bra66 sectVII] and [CG12 Theorem 53] for these results


Lemma 25 Let G = PGL2(q) for a prime power q such that the Sylow 2-subgroupsof G are dihedral of order at least 8 and let H le G be a subgroup isomorphic to PSL2(q)Moreover let Q isin Syl2(H) and P isin Syl2(G) such that P cap H = Q Without loss of

generality we may set P = 〈s t | s2nminus1

= t2 = 1 tst = sminus1〉 and Q = 〈s2 t〉 Then thefollowing holds

(a) st is an involution in P Q and moreover any two involutions in P Q areP -conjugate

(b) Set z = s2nminus2

Then centralisers of involutions in P and G are given as follows

(i) CP (z) = P CP (t) = 〈t z〉 sim= C2 times C2 and CP (st) = 〈st z〉 sim= C2 times C2

(ii) If q equiv 1 (mod 4) then CG(t) sim= D2(qminus1) and CG(st) sim= D2(q+1)

(iii) If q equiv minus1 (mod 4) then CG(t) sim= D2(q+1) and CG(st) sim= D2(qminus1)

In particular CP (z) isin Syl2(CG(z)) and CP (st) isin Syl2(CG(st))

Proof By assumption |G H| = 2 P sim= D2n with n ge 3 and Q sim= D2nminus1 The threeP -conjugacy classes of involutions in P are

s2nminus2

s2jt | 0 le j le 2nminus2 and s2j+1t | 0 le j le 2nminus2

where s2nminus2

s2jt | 0 le j le 2nminus2 sub Q and s2j+1t | 0 le j le 2nminus2 sub P Q Part (a)follows

For part (b) (i) is obvious and for (ii) and (iii) we refer to [GW62 sect4(B)] for the descrip-tion of centralisers of involutions in PGL2(q) Now it is clear that CP (z) isin Syl2(CG(z))If q equiv 1 (mod 4) then 2 || q + 1 so that |CG(st)|2 = |D2(q+1)|2 = 4 = |CP (st)| whereasif q equiv minus1 (mod 4) then 2 || q minus 1 so that |CG(st)|2 = |D2(qminus1)|2 = 4 = |CP (st)| HenceCP (st) isin Syl2(CG(st))

3 Properties of Scott modules

Lemma 31 Let G and Gprime be finite p-nilpotent groups with a common Sylow p-subgroup P Then Sc(GtimesGprime ∆P ) induces a Morita equivalence between B0(kG) and B0(kG

prime)

Proof Set M = Sc(G times Gprime ∆P ) B = B0(kG) and Bprime = B0(kGprime) By definition we

have

M∣∣∣ 1B middot kGotimeskP kGprime middot 1Bprime

Since G and Gprime are p-nilpotent 1B middot kG otimeskP kGprime middot 1Bprime = kP otimeskP kP sim= kP as (kG kGprime)-bimodules Now as kP is indecomposable as a (kP kP )-bimodule it is also indecom-posable as a (kG kGprime)-bimodule Therefore M sim= kP as (BBprime)-bimodule Therefore Minduces a Morita equivalence between B and Bprime

Lemma 32 Assume that p = 2 Let G and Gprime be two finite groups with a common Sylow2-subgroup P and assume that FP (G) = FP (G

prime) Let z be an involution in Z(P ) Then

Sc(CG(z)times CGprime(z) ∆P

) ∣∣∣(Sc(GtimesGprime ∆P )

)(∆〈z〉)

Proof Since FP (G) = FP (Gprime) we have F∆P (G times Gprime) sim= FP (G) Since P is a Sylow

2-subgroup of G FP (G) is saturated and therefore F∆P (GtimesGprime) is also saturated Since∆〈z〉 le Z(∆P ) ∆〈z〉 is a fully normalised subgroup of ∆P with respect to F∆P (GtimesGprime)by definition Thus it follows from [IK17 Lemmas 31 and 22] that

Sc(NGtimesGprime(∆〈z〉) N∆P (∆〈z〉))∣∣∣Sc(GtimesGprime ∆P )darrNGtimesGprime (∆〈z〉)


Since z is an involution in Z(P ) the above reads

Sc(CG(z)times CGprime(z) ∆P )∣∣∣Sc(GtimesGprime ∆P )darrCG(z)timesCGprime (z)

Set M = Sc(G times Gprime ∆P ) and M = Sc(CG(z) times CGprime(z) ∆P ) Since taking the Brauerconstruction is functorial by the above we have that

M(∆〈z〉)∣∣∣M(∆〈z〉)

Obviously ∆〈z〉 is in the kernel of the (kCG(z)times kCGprime(z))-bimodule kCG(z)otimeskP kCGprime(z)and hence it is in the kernel of M so that M∆〈z〉 = M Therefore by definition of theBrauer construction we have M(∆〈z〉) = M Therefore M | M(∆〈z〉)

The following is well-known but does not seem to appear in the literature For com-pleteness we provide a proof which is due to N Kunugi

Lemma 33 Let G and Gprime be finite groups with a common Sylow p-subgroup P suchthat FP (G) = FP (G

prime) Assume further that M is an indecomposable ∆P -projective p-permutation k(GtimesGprime)-module Let Q be a subgroup of P Then the following are equiv-alent

(i) Sc(Gprime Q) | (kG otimeskG M)(ii) Sc(GtimesGprime ∆Q) |M

Proof By assumption M | k∆RuarrGtimesGprime

where ∆R is a vertex of M with R le P Set

N = k∆QuarrGtimesGprime

= kGotimeskQ kGprime Thus N = Sc(GtimesGprime ∆Q)oplusN0 where N0 is a k(GtimesGprime)-module which satisfies Homk(GtimesGprime)(N0 kGtimesGprime) = 0 by definition of a Scott module Wehave that

kG otimeskG N = kG otimeskG (kGotimeskQ kGprime)

= kGdarrQuarrGprime

= kQuarrGprime

= Sc(Gprime Q)oplusX

where X is a kGprime-module such that no Scott module can occur as a direct summand ofX thanks to Frobenius Reciprocity Now let us consider an arbitrary k(GtimesGprime)-moduleL with L |N Then

dim[HomkGprime(kG otimeskG L kGprime)] = dim[Homk(GtimesGprime)(L kG otimesk kGprime)]

= dim[Homk(GtimesGprime)(L kGtimesGprime)]

le dim[HomGtimesGprime(N kGtimesGprime)] = 1

by adjointness and Frobenius Reciprocity Hence we have

dim[Homk(GtimesGprime)(L kGtimesGprime)] =

1 if Sc(GtimesGprime ∆Q) |L

0 otherwise

Namely

Sc(Gprime Q) | (kG otimeskG L) if and only if Sc(GtimesGprime ∆Q) |L

Now we can decompose M into a direct sum of indecomposable k(GtimesGprime)-modules

M = M1 oplus middot middot middot oplusMs oplus Y1 oplus middot middot middot oplus Yt


for integers s ge 1 and t ge 0 and where for each 1 le i le s Mi = Sc(G times Gprime ∆Qi) forsome Qi le P and for each 1 le j le t Yj 6sim=Sc(GtimesGprime ∆S) for any S le P Then for each1 le i le s we have

kG otimeskG Mi = kG otimeskG Sc(GtimesGprime ∆Qi)∣∣∣ kG otimeskG (kGotimeskQi

kGprime)

where

kG otimeskG (kGotimeskQikGprime) = kGdarrQi

uarrGprime

= kQiuarrG

prime

= Sc(Gprime Qi)oplusWi

for a kGprime-module Wi such that HomkGprime(kGprime Wi) = 0 by Frobenius Reciprocity NowFP (G) = FP (G

prime) implies that FP (G) = FP (Gprime) sim= F∆P (G times Gprime) This means that for

1 le j jprime le s we have that Qj is Gprime-conjugate to Qjprime if and only if ∆Qj is (GtimesGprime)-conjugate to ∆Qjprime Therefore the claim follows directly from the characterisation of Scottmodules in [NT88 Corollary 485]

Lemma 34 Let G and Gprime be finite groups with a common Sylow p-subgroup P such thatFP (G) = FP (G

prime) Set M = Sc(G times Gprime ∆P ) B = B0(kG) and Bprime = B0(kGprime) If M

induces a stable equivalence of Morita type between B and Bprime then the following holds

(a) kG otimesB M = kGprime(b) If U is an indecomposable p-permutation kG-module in B with vertex 1 6=Q le P

then UotimesBM has up to isomorphism a unique indecomposable direct summand Vwith vertex Q and V is again a p-permutation module

(c) For any Q le P Sc(G Q)otimesB M = Sc(Gprime Q)oplus (proj)(d) For any Q le P ΩQ(kG)otimesB M = ΩQ(kGprime)oplus (proj)

Proof (a) Apply Lemma 33 to the case that Q = P Then Condition (ii) is triviallysatisfied Thus we have kGprime | (kG otimesB M) since Sc(Gprime P ) = kGprime Hence Theorem 21(a)yields kGprime = kG otimesB M

(b) Let U be an indecomposable p-permutation kG-module with vertex Q Since Minduces a stable equivalence of Morita type between B and Bprime U otimesB M has a uniquenon-projective indecomposable direct summand say V Then

V∣∣∣ (U otimeskG M)

∣∣∣ [U otimeskG (kGotimeskP kGprime)]

= UdarrPuarrGprime

∣∣∣ kQuarrGdarrPuarrGprime

=oplus

gisin[QGP ]

kQgcapPuarrGprime

by the Mackey decomposition Hence V is a p-permutation kGprime-module which is (Qg cap P )-projective for an element g isin G Thus there is a vertex R of V with R le Qg cap P Thismeans that gRgminus1 le Q cap gPgminus1 le Q le P Then since FP (G) = FP (G

prime) there is anelement gprime isin Gprime such that grgminus1 = gprimergprimeminus1 for every r isin R Hence gprimeRgprimeminus1 is also a vertexof V and gprimeRgprimeminus1 = gRgminus1 le Q and hence R leGprime Q

Similarly we obtain that Q leG R since Mlowast induces a stable equivalence of Morita typebetween Bprime and B This implies that R =Gprime Q and R =G Q


(c) Set F = minus otimesB M the functor inducing the stable equivalence of Morita typebetween B and Bprime Fix Q with 1 6=Q le P and set UQ = Sc(GQ) Then we have

1 = dim[HomB(UQ kG)]

= dim[HomB(UQ kG)]

= dim[HomBprime

(F (UQ) F (kG)

)]

= dim[HomBprime

(F (UQ) kGprime

)]

= dim[HomBprime

(F (UQ) kGprime

)]

where the second equality holds because kG is simple and the last but one equality holdsby (a) Let VQ be the unique (up to isomorphism) non-projective indecomposable directsummand of F (UQ) By (b) we know that Q is a vertex of VQ Moreover by the abovewe have that

dim[HomBprime(VQ kGprime)] le 1

We claim that in fact equality holds Indeed if dim[HomBprime(VQ kGprime)] = 0 then the aboveargument implies that

dim[HomkG(Fminus1(VQ) kG)] = 0

as well but this is a contradiction since UQ is a direct summand of Fminus1(VQ) as thefunctor F gives a stable equivalence between B and Bprime Hence the dimension is one andwe conclude that VQ = Sc(Gprime Q)

Next assume that Q = 1 so that Sc(GQ) = Sc(G 1) = P (kG) Since

P (kG)otimeskG M∣∣∣P (kG)otimeskG kGotimeskP kGprime = P (kG)darrPuarr

Gprime

by definition of a Scott module P (kG) otimeskG M is a projective kGprime-module Moreover itfollows from the adjointness and (a) that

dim[HomkGprime(P (kG)otimeskG M kGprime)] = dim[HomkG(P (kG) kGprime otimeskGprime Mlowast)] = 1

and hence P (kG)otimeskG M = P (kGprime)oplus P where P is a projective Bprime-module that does nothave P (kGprime) as a direct summand

(d) Recall that Sc(GQ) = PQ(kG) Sc(Gprime Q) = PQ(kGprime) and ΩQ(kG) = PQ(kG)kG and

ΩQ(kGprime) = PQ(kGprime)kGprime Therefore (d) follows directly from (a) (c) and the stripping-offmethod [KMN11 (A1) Lemma]

4 Constructing stable equivalences of Morita type

We start with the following gluing result which will allow us to construct stable equiv-alences of Morita type It is essentially due to Broue ([Bro94 63Theorem]) Rouquier([Rou01 Theorem 56]) and Linckelmann ([Lin01 Theorem 31]) We aim at using equiv-alence (iii) which slightly generalises the statement of [Rou01 Theorem 56] Since astatement under our hypotheses does not seem to appear in the literature we give a prooffor completeness

Lemma 41 Let G and Gprime be finite groups with a common Sylow p-subgroup P andassume that FP (G) = FP (G

prime) Set M = Sc(G times Gprime ∆P ) B = B0(kG) and Bprime =B0(kG

prime) Further for each subgroup Q le P we set BQ = B0(kCG(Q)) and BprimeQ =

B0(kCGprime(Q)) Then the following three conditions are equivalent

(i) The pair (M Mlowast) induces a stable equivalence of Morita type between B and Bprime


(ii) For every non-trivial subgroup Q le P the pair (M(∆Q) M(∆Q)lowast) induces aMorita equivalence between BQ and Bprime

Q(iii) For every cyclic subgroup Q le P of order p the pair (M(∆Q) M(∆Q)lowast) induces

a Morita equivalence between BQ and BprimeQ

Proof (i)hArr(ii) is a special case of [Lin01 Theorem 31] and (ii)rArr(iii) is trivial

(iii)rArr(ii) Take an arbitrary non-trivial subgroup Q le P Then there is a normal series

Cpsim= Q1 ⊳ Q2 ⊳ middot middot middot ⊳ Qm = Q

for an integer m ge 1 We shall prove (ii) by induction on m If m = 1 then M(∆Q)induces a Morita equivalence between BQ and Bprime

Q by (iii)Next assume that m ge 2 and (ii) holds for mminus 1 and set R = Qmminus1 Namely by the

inductive hypothesis M(∆R) realises a Morita equivalence between BR and BprimeR That is

to say we have

(1) BRsim= M(∆R)otimesBprime

RM(∆R)lowast

Moreover

(2) BR(∆Q) = BQ

since (kCG(R))(∆Q) = kCG(Q) (note that since R ⊳ Q kCG(R) is a right k∆Q-module)Further it follows from [Ric96 proof of Theorem 41] that

(3)(M(∆R) otimesBprime

RM(∆R)lowast

)(∆Q) = (M(∆R)) (∆Q)otimesBprime

Q[(M(∆R)) (∆Q)]lowast

Recall that by [BP80 Proposition 15] we have

(4) (M(∆R)) (∆Q) = M(∆Q)

Hence

BQ = (BR)(∆Q) by (2)

=(M(∆R)otimesBprime

RM(∆R)lowast

)(∆Q) by (1)

= (M(∆R)) (∆Q)otimesBprime

Q[(M(∆R)) (∆Q)]lowast by (3)

= M(∆Q)otimesBprime

QM(∆Q)lowast by (4)

Thus by making use of [Ric96 Theorem 21] we obtain that the pair (M(∆Q) M(∆Q)lowast)induces a Morita equivalence between BQ and Bprime

Q

From now on and until the end of this article we assume that k has characteristic 2

The following is an easy application of the Baer-Suzuki theorem which is essential totreat dihedral defect groups

Lemma 42 Let G be a finite group and let Q be a normal 2-subgroup of G such thatGQ sim= S3 Assume further that there is an involution t isin GQ Then G has a subgroupH such that t isin H sim= S3


Proof Obviously Q = O2(G) since GQ sim= S3 Therefore by the Baer-Suzuki theorem(see [Gor68 Theorem 382]) there exists an element y isin G such that y is conjugate to

t in G and the group H = 〈t y〉 is not a 2-group Therefore 6 | |H| and since H isgenerated by two involutions it is a dihedral group of order 3 middot2a for some positive integer

a that is H = C3middot2aminus1 ⋊ 〈t〉 (see [Gor68 Theorem 911]) Seeing H as generated by yt

and t it follows immediately that H has a dihedral subgroup of order 6 say H generatedby t and a suitable power of yt The claim follows

Corollary 43 Let G be a finite group with a dihedral 2-subgroup P of order at least 8and let Q P such that Q sim= C2 times C2 Assume moreover that CG(Q) is 2-nilpotent andthat NG(Q)CG(Q) sim= S3 Then there exists a subgroup H of NG(Q) such that NP (Q) isa Sylow 2-subgroup of H and |NG(Q) H| is a power of 2 (possibly 1)

Proof Let K = O2prime(CG(Q)) and let R isin Syl2(CG(Q)) Then by assumption we have thefollowing inclusions of subgroups

NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R

K timesQ = K ⋊ (C2 times C2)

where we note that |NP (Q)Q| = 2 by [Bra74 Proposition (1B)] hence K ⋊ NP (Q) =K ⋊D8 Since [K Q] = 1 by the choice of K we have K ⋊Q = K timesQ Furthermore asK is a characteristic subgroup of CG(Q) and CG(Q) ⊳ NG(Q) we have K ⊳ NG(Q) sothat (K times Q) ⊳ NG(Q) Clearly (K times Q) ⊳ (K ⋊ NP (Q)) since the index is two AlsoCG(Q) cap (K ⋊NP (Q)) = K timesQ Therefore we can take quotients by L = K times Q of allthe groups in the picture This yields the following inclusions of subgroups

G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L

〈1〉 = (K timesQ)L = Q cap R

In particular we have that GQ sim= S3 R = (K ⋊ NP (Q))L sim= NP (Q)Q sim= C2 Q =(K⋊R)(KtimesQ) sim= RQ (which is a 2-group) and QcapR = 〈1〉 Now there must exist aninvolution t isin R such that t 6isin Q Therefore by Lemma 42 there exists a subgroup H of


G such that t isin H sim= S3 Finally we set H to be the preimage of H under the canonicalhomomorphism NG(Q) NG(Q)L and the claim follows

We can now prove Theorem 13 of the Introduction

Proof of Theorem 13 Set M = Sc(GP ) Let Q le P be an arbitrary fully normalisedsubgroup in FP (G) We claim that if Q 6= 1 then NG(Q) has a subgroup HQ whichsatisfies the conditions (1) and (2) in Theorem 24

First suppose that Q 6sim=C2 times C2 Then Aut(Q) is a 2-group (see [Gor68 Lemma 541(i)-(ii)]) and hence NG(Q)CG(Q) is also a 2-group Thus NG(Q) is 2-nilpotent sinceCG(Q) is 2-nilpotent by the assumption Set N = NG(Q) and hence we can writeN = K ⋊ PN where K = O2prime(N) and PN is a Sylow 2-subgroup of N Since NP (Q) is a2-subgroup of N we can assume PN ge NP (Q) Set HQ = K ⋊NP (Q) Then obviouslyNP (Q) is a Sylow 2-subgroup of HQ and |N HQ| is a power of 2 (possibly one) since|N HQ| = |PN NP (Q)| This means that Q satisfies the conditions (1) and (2) inTheorem 24

Next suppose that Q sim= C2 times C2 Then NG(Q)CG(Q) rarr Aut(Q) sim= S3 ClearlyCP (Q) = Q and NP (Q) sim= D8 (see [Bra74 Proposition (1B)]) Then as NP (Q)CP (Q) rarrNG(Q)CG(Q) we have that |NG(Q)CG(Q)| isin 2 6 If |NG(Q)CG(Q)| = 2 thenNG(Q) is 2-nilpotent since CG(Q) is 2-nilpotent so that using an argument similar to theone in the previous case there exists a subgroup HQ of NG(Q) such that HQ satisfies theconditions (1) and (2) in Theorem 24 Hence we can assume that |NG(Q)CG(Q)| 6=2Now assume that |NG(Q)CG(Q)| = 6 so that NG(Q)CG(Q) sim= S3 Then it follows fromCorollary 43 that NG(Q) has a subgroup HQ such that NP (Q) is a Sylow 2-subgroup ofHQ and that |NG(Q) HQ| is a power of 2 as required Therefore by Theorem 24

Sc(NG(Q) NP (Q)) darrNG(Q)QCG(Q) is indecomposable for each fully normalised subgroup 1 6=

Q le P

Now if Q = 1 then Sc(NG(Q) NP (Q))darrNG(Q)QCG(Q)= Sc(GP ) which is indecomposable by

definitionTherefore Theorem 23 yields that M is Brauer indecomposable

Corollary 44 Let G and Gprime be finite groups with a common Sylow 2-subgroup P whichis a dihedral group of order at least 8 and assume that FP (G) = FP (G

prime) Then the Scottmodule Sc(GtimesGprime∆P ) is Brauer indecomposable

Proof Set M = Sc(GtimesGprime∆P ) Since P is a Sylow 2-subgroup of G FP (G) is saturatedby [BLO03 Proposition 13] Therefore as F∆P (GtimesGprime) sim= FP (G) by definition we havethat F∆P (GtimesGprime) is saturated

Now let Q le ∆P be any fully normalised subgroup in F∆P (G times Gprime) Obviously wecan write Q = ∆Q for a subgroup Q le P We claim that the Brauer constructionM(∆Q) is indecomposable as a k[CGtimesGprime(∆Q)]-module Notice that clearly CGtimesGprime(∆Q) =CG(Q)times CGprime(Q)

If Q = 1 then the claim is obvious since M(∆〈1〉) = M So assume that Q 6=1 HenceQ contains an involution t Thus CG(Q) le CG(t) so that [Bra66 Lemma (7A)] impliesthat CG(Q) is 2-nilpotent and similarly for CGprime(Q) Hence CGtimesGprime(∆Q) is 2-nilpotentTherefore it follows from Theorem 13 that M is Brauer indecomposable

Lemma 45 Assume that G and Gprime are finite groups with a common Sylow 2-subgroup Pwhich is a dihedral group of order at least 8 Assume moreover that FP (G) = FP (G

prime)and is such that there are exactly two G-conjugacy classes of involutions in P Further


suppose that z isin Z(P ) and t isin P are two involutions in P which are not G-conjugateSet M = Sc(GtimesGprime ∆P ) Then

M(∆〈t〉) sim= Sc(CG(t)times CGprime(t) ∆CP (t))

as k[CG(t)times CGprime(t)]-modules

Proof Set F = F∆P (GtimesGprime) By the beginning of the proof of Lemma 32 F is saturatedSet Q = 〈t〉

Next we claim that Q is a fully FP (G)-normalised subgroup of P Let R le P beFP (G)-conjugate to Q So that R and Q are G-conjugate So we can write R = 〈r〉 foran element r isin R since |Q| = 2 Thus t and r are G-conjugate which implies that r andz are not G-conjutate by the assumption and hence r and t are P -conjugate again bythe assumption Namely t = rπ for an element π isin P Obviously CP (r)

π = CPπ(rπ) =CP (r

π) = CP (t) so that |CP (r)| = |CP (t)| which yields that |NP (R)| = |NP (Q)| since|R| = |Q| = 2 Therefore Q is a fully FP (G)-noramlised subgroup of P

Thus ∆Q is a fully F -normalised subgroup of ∆P since F sim= FP (G) = FP (Gprime) (see

the beginning of the proof of Lemma 32) Moreover M is Brauer indecomposable byLemma 44 Hence it follows from [IK17 Theorem 31] that

M(∆Q) sim= Sc(NGtimesGprime(∆Q) N∆P (∆Q))

as k[NGtimesGprime(∆Q)]-modules Noting that |∆Q| = 2 and ∆Q = ∆〈t〉 we have that

NGtimesGprime(∆Q) = CGtimesGprime(∆Q) = CG(Q)times CGprime(Q) = CG(t)times CGprime(t)

and thatN∆P (∆Q) = C∆P (∆Q) = ∆CP (Q) = ∆CP (t)

The assertion follows

Proposition 46 Let G and Gprime be finite groups with a common Sylow 2-subgroup Pwhich is a dihedral group of order at least 8 Assume moreover that O2prime(G) = O2prime(G

prime) = 1and FP (G) = FP (G

prime) Then the Scott module Sc(GtimesGprime∆P ) induces a stable equivalenceof Morita type between the principal blocks B0(kG) and B0(kG

prime)

Proof Fix P = D2n for an n ge 3 and set M = Sc(G times Gprime∆P ) B = B0(kG) andBprime = B0(kG

prime) Since we assume that O2prime(G) = O2prime(Gprime) = 1 G and Gprime are amongst the

groups (D1)-(D3) listed in sect24 First we note that G = P is the unique group in thislist with FP (G) = FP (P ) therefore we may assume that G 6= P 6= Gprime Thus by sect25and by the assumption that FP (G) = FP (G

prime) we have that P has the same number ofG-conjugacy and Gprime-conjugacy classes of involutions namely either one or two

Let t isin P be an arbitrary involution and set Bt = B0(kCG(t)) and Bprimet = B0(kCGprime(t))

We claim that M(∆〈t〉) induces a Morita equivalence between Bt and Bprimet

First of all assume that t isin Z(P ) Set z = t Thus CP (z) = P and this is a Sylow2-subgroup of CG(z) so that P is a Sylow 2-subgroup of CGprime(z) as well Recall that CG(t)and CGprime(t) are both 2-nilpotent by [Bra66 Lemma (7A)] Set

Mz = Sc(CG(z)times CGprime(z) ∆P )

By Lemma 31 Mz induces a Morita equivalence between Bz and Bprimez Hence we have

Mz |M(∆〈z〉) by Lemma 32 and Corollary 44 yields

M(∆〈z〉) = Mz

This means that M(∆〈z〉) induces a Morita equivalence between Bz and Bprimez


Case 1 Assume first that all involutions in P are G-conjugate so that all involutionsin P are Gprime-conjugate a well since FP (G) = FP (G

prime) Therefore there exists an elementg isin G and an element gprime isin Gprime such that zg = t = zg

prime

Thus by definition of the Brauerconstruction we have

M(∆〈t〉) = M(∆〈zg〉) = M((∆〈z〉)(ggprime)) = M(∆〈z〉)(gg

prime) = (Mz)(ggprime)

Moreover we have

(Mz)(ggprime) = (Sc(CG(z)times CGprime(z) ∆P ))(gg

prime)

= Sc(CG(zg)times CGprime(zg

prime

) (∆P )(ggprime))

= Sc(CG(t)times CGprime(t) ∆)

where ∆ = (πg πgprime) | π isin P sim= P Obviously P g is a Sylow 2-subgroup of CG(t) andP gprime is a Sylow 2-subgroup of CGprime(t) Further CG(t) and CGprime(t) are 2-nilpotent Thereforeit follows from Lemma 31 that

Sc(CG(t)times CGprime(t) ∆) = M(∆〈t〉)

induces a Morita equivalence between Bt and Bprimet

Case 2 Assume now that P has exactly two G-conjugacy classes and hence exactly twoGprime-conjugacy classes of involutions Then by sect25 G and Gprime are groups of type (D3)(ii)that is G sim= PGL2(q)⋊Cf and Gprime sim= PGL2(q

prime)⋊Cf prime for some odd prime powers q qprime andsome suitable odd positive integers f f prime

If t is G-conjugate to the central element z isin Z(P ) then M(∆〈t〉) induces a Moritaequivalence between Bt and Bprime

t by the same argument as in Case 1 Hence we may assumethat t is not G-conjugate to z We note that by Lemma 25(a) any two involutions in Pwhich are not G-conjugate (resp Gprime-conjugate) to z are already P -conjugate It followseasily from Lemma 25(b) that CP (t) is a Sylow 2-subgroup of both CG(t) and CG(t

prime)Again because CG(t) and CGprime(t) are both 2-nilpotent it follows from Lemma 31 that

Mt = Sc(CG(t)times CGprime(t) ∆CP (t))

induces a stable equivalence of Morita type between B and BprimeOn the other hand Lemma 45 implies that

M(∆〈t〉) = Mt

Therefore M(∆〈t〉) induces a Morita equivalence between Bt and Bprimet Hence the claim

holdsFinally Lemma 41 yields that M induces a stable equivalence of Morita type between

B and Bprime

Corollary 47 Let G and Gprime be two finite groups with a common Sylow 2-subgroupP sim= D2n with n ge 3 and let M = Sc(GtimesGprime∆P )

(a) If G = PSL2(q) and Gprime = PSL2(qprime) where q and qprime are powers of odd primes such

that q equiv qprime (mod 8) and |G|2 = |Gprime|2 ge 8 then M induces a stable equivalence ofMorita type between B0(kG) and B0(kG

prime)(b) If n = 3 G = A7 and Gprime = PSL2(q

prime) where qprime is a power of an odd prime suchthat |Gprime|2 = 8 then M induces a stable equivalence of Morita type between B0(kG)and B0(kG

prime)


(c) If G = PGL2(q) and Gprime = PGL2(qprime) where q and qprime are powers of odd primes

such that q equiv qprime (mod 4) and |G|2 = |Gprime|2 then M induces a stable equivalence ofMorita type between B0(kG) and B0(kG

prime)

Furthermore there exists a stable equivalence of Morita type between B0(kG) and B0(kGprime)

if and only if FP (G) = FP (Gprime)

Proof Parts (a) (b) and (c) follow directly from Proposition 46 since in each caseFP (G) = FP (G

prime) The sufficient condition of the last statement also follows from Proposi-tion 46 since we have already noticed that inflation induces splendid Morita equivalences(hence stable equivalences of Morita type) between B0(kG) and B0(kGO2prime(G)) respbetween B0(kG

prime) and B0(kGO2prime(Gprime)) To prove the necessary condition we recall that

the existence of a stable equivalence of Morita type between B0(kG) and B0(kGprime) implies

that

k(B0(G))minus l(B0(G)) = k(B0(Gprime))minus l(B0(G

prime))

(see [Bro94 53Proposition]) Using [Bra66 Theorem (7B)] we have that k(B0(G)) =k(B0(G

prime)) = 2nminus2 + 3 so that we must have l(B0(G)) = l(B0(Gprime)) which in turn forces

FP (G) = FP (Gprime) see sect25

5 The principal blocks of PSL2(q) and PGL2(q)

Throughout this section we assume that k is a field of characteristic 2 We now start todetermine when the stable equivalences of Morita type we constructed in the previous sec-tion are actually Morita equivalences and in consequence splendid Morita equivalencesWe note that these Morita equivalences are known from the work of Erdmann [Erd90](over k) or Plesken [Ple83 VII] (over complete discrete valuation rings) but the methodsused do not prove that they are splendid Morita equivalences

We start with the case PSL2(q) and we fix the following notation we set B(q) =CZ(SL2(q)) where C le SL2(q) is the subgroup of upper-triangular matrices We have|PSL2(q)| =

12q(qminus1)(q+1) and |B(q)| = 1

2q(qminus1) Furthermore the principal 2-block of

PSL2(q) contains three simple modules namely the trivial module and two mutually dualmodules of k-dimension (q minus 1)2 which we denote by S(q) and S(q)lowast (See eg [Erd77Lemma 43 and Corollary 52]) Throughout we heavily rely heavily on Erdmannrsquos com-putation of the PIMs of PSL2(q) in [Erd77]

Lemma 51 Let G = PSL2(q) where q is a power of an odd prime and let P be aSylow 2-subgroup of G Then the Loewy and socle series structure of the Scott modulewith respect to B(q) is

Sc(G B(q)) =kG

S(q)oplus S(q)lowast

kG

Before proceeding with the proof we note that in this lemma we allow the Sylow 2-subgroups to be Klein-four groups as this case will be necessary when dealing with thegroups of type PGL2(q) and dihedral Sylow 2-subgroups of order 8

Proof Assume first that q equiv minus1 (mod 4) Then 2 ∤ |B(q)| so that Sc(GB(q)) = P (kG)(see [NT88 Corollary 485]) Therefore by [Erd77 Theorem 4(a)] the Loewy and soclestructure of the PIM P (kG) = Sc(GB(q)) is as claimed


Assume next that q equiv 1 (mod 4) By [Bon11 sect323] the trivial source module kB(q)uarrG

affords the ordinary character

(5) 1B(q)uarrG = 1G + StG

where StG denotes the Steinberg character Therefore kB(q)uarrG is indecomposable and

isomorphic to Sc(G B(q)) = X Then it follows from [Bon11 Table 91] that

X = kG + (kG + S(q) + S(q)lowast)

as composition factors Since X is an indecomposable self-dual 2-permutation kG-moduleits Loewy and socle structure is one of

kGS(q)oplus S(q)lowast

kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG

kG oplus S(q)kG oplus S(q)lowast

orkG oplus S(q)lowast

kG oplus S(q)

By [Erd77 Theorem 2(a)] Ext1kG(S(q) S(q)lowast) = 0 = Ext1kG(S(q)

lowast S(q)) hence the secondand the third cases cannot occur

Suppose now that the fourth case happens Then X has a submodule Y such thatXY sim= kG Hence Y = S(q) + S(q)lowast + kG as composition factors Then sinceExt1kG(S(q) S(q)

lowast) = 0 Y has the following structure

Y =S(q)kG

oplus S(q)lowast

Hence Y has a submodule Z with the Loewy and socle structure

Z =S(q)kG

Similarly X has a submodule W such that XW sim= S(q) and W = kG + kG + S(q)lowast ascomposition factors By [Erd77 Theorem 2] Ext1kG(kG kG) = 0 Therefore W has thefollowing structure

W =kG

S(q)lowastoplus kG

and hence W has a submodule U with structure

U =kG

S(q)lowast

Since Z and U are submodules of X and Z cap U = 0 we have a direct sum Z oplus U in X As a consequence X = ZoplusU which is a contradiction since X is indecomposable Hencethe fourth case cannot occur

Similarly the fifth case cannot happen Therefore we must have that

Sc(G B(q)) =kG


kG

as desired


Proposition 52 Let G = PSL2(q) and Gprime = PSL2(qprime) where q and qprime are powers of

odd primes such that q equiv qprime (mod 4) and |G|2 = |Gprime|2 ge 8 Let P be a common Sylow2-subgroup of G and Gprime Then the Scott module Sc(GtimesGprime ∆P ) induces a splendid Moritaequivalence between B0(kG)and B0(kG

prime)

Proof Set M = Sc(G times Gprime∆P ) and B = B0(kG) and Bprime = B0(kGprime) First by

Proposition 47(a) M induces a stable equivalence of Morita type between B and Bprime Weclaim that this is a Morita equivalence Using Theorem 21(b) it is enough to check thatthe simple B-modules are mapped to the simple Bprime-modules

To start with by Proposition 47(a) and Lemma 34(a) we have

(6) kG otimesB M = kGprime

Next because q equiv qprime (mod 4) the Scott modules Sc(G B(q)) and Sc(Gprime B(qprime)) have acommon vertex Q (which depend on the value of q modulo 4) Therefore by Lemma 34(c)we have

Sc(G B(q))otimesB M = Sc(Gprime B(qprime))oplus (proj)

Moreover as Sc(G B(q)) and Sc(Gprime B(qprime)) are the relative Q-projective covers of kGand kGprime respectively it follows from Lemma 51 that the socle series of ΩQ(kG) andΩQ(kGprime) are given by

ΩQ(kG) =S(q)oplus S(q)lowast

kGand ΩQ(kGprime) =

S(qprime)oplus S(qprime)lowast

kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)

Then it follows from (6) and [KMN11 Lemma A1] (stripping-off method) that

(S(q)otimesB M)oplus (S(q)lowast otimesB M) = (S(q)oplus S(q)lowast)otimesB M = S(qprime)oplus S(qprime)lowastoplus (proj)

Thus Theorem 21(a) implies that both (non-projective) simple B-modules S(q) and S(q)lowast

are mapped to a simple Bprime-moduleIn conclusion Theorem 21(b) yields that M induces a Morita equivalence As M is

a 2-permutation k(G times Gprime)-module the Morita equivalence induced by M is actually asplendid Morita equivalence see sect22

Next we consider the case PGL2(q) We fix a subgroup H(q) lt PGL2(q) such thatH(q) sim= PSL2(q) and keep the notation B(q) lt H(q) as above Furthermore the principal2-block of PGL2(q) contains two simple modules namely the trivial module and a self-dualmodule of dimension q minus 1 which we denote by T (q)

Lemma 53 Let G = PGL2(q) where q is a power of an odd prime Then the Loewyand socle sturucture of the Scott module Sc(G B(q)) with respect to B(q) is as follows

Sc(G B(q)) =

kG

kG oplus T (q)T (q)oplus kG

kG


Proof Write H = H(q) B = B(q) Y = Sc(H B) and X = Y uarrG Since |GH| = 2Greenrsquos indecomposability theorem and Frobenius Reciprocity imply that X = Sc(G B)

Let χ be the ordinary character of G afforded by the 2-permutation kG-module X Itfollows from equation (5) [Bon11 Table 91] and Clifford theory that

(7) χ = 1BuarrHuarrG = 1G + 1prime + χSt1 + χSt2

where 1G is the trivial character 1prime is the non-trivial linear character of G and χSti fori = 1 2 are the two distinct irreducible constituents of χSt uarr

G of degree q Using [Bon11Table 91] we have that the 2-modular reduction of X is

X = kG + kG + (kG + T (q)) + (kG + T (q))

as composition factors Moroeover by Proposition 51 we have that the Loewy and socleseries of Y is

Y =kH


kH

Assume first that q equiv minus1 (mod 4) As in the proof of Lemma 51 as 2 ∤ B| we havethat

P (kG) = Sc(GB)

Moreover by Webbrsquos theorem [Web82 Theorem E] the heart H(P (kG)) of P (kG) isdecomposable with precisely two indecomposable summands and by [AC86 Lemma 54and Theorem 55] these two summands are dual to each other and endo-trivial modulesIt follows that P (kG) must have the following Loewy and socle structure

Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG

Next assume that q equiv 1 (mod 4) Since X = Y uarrG T (q) = S(q)uarrG = S(q)lowastuarrG Frobe-nius Reciprocity implies that X(X rad(kG)) sim= soc(X) sim= kG and that X has a filtrationby submodules such that X 13 X1 13 X2 such that

XX1sim=

kGkG

X1X2sim= T (q)oplus T (q) and X2

sim=kGkG

by Proposition 51 Since X is a 2-permutation kG-module we know by (7) and Scottrsquostheorem on the lifting of homomorphisms (see [Lan83 Theorem II124(iii)]) that

dim[HomkG(XkGkG

)] = 2

This implies that X has both a factor module and a submodule which have Loewy struc-ture

kGkG


Now we note that kG occurs exactly once in the second Loewy layer ofX as |GO2(G)| = 2and we have that the Loewy structure of X is of the form

X =

kGkG middot middot middot

kG

so that only kG+(2timesT (q)) are left to determine Further we know from Shapirorsquos Lemmaand [Erd77 Theorem 2] that Ext1kG(T (q) T (q)) = 0 and that

dim[Ext1kG(kG T (q))] = dim[Ext1kG(T (q) kG)] = 1

Hence using the above filtration of X we obtain that the Loewy structure of XX2 is

(8) XX2 =kG

kG oplus T (q)T (q)

Thus X has Loewy structure

(9)

kGkG oplus T (q)T (q)oplus kG

kG

or

kGkG oplus T (q)

T (q)kGkG

It follows from (8) and the self-dualities of kG T (q) and X that X has a submodule X3

such that the socle series has the form

(10) X3 =T (q)

kG oplus T (q)kG

(socle series)

First assume that the second case in (9) holds Then again by the self-dualities thesocle series of X has the form

(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)

By making use of (9) and (11) we have(12)

X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)


It follows from (12) that (up to isomorphism) there are exactly four factor modulesU1 U4 of X such that Ui(Ui rad(kG)) sim= soc(Ui) sim= kG for each i and furthermorethat these have structures such that

(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG

Now we know by (12) that X has the following socle series

(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)

If X has a submodule isomorphic to U4 then this contradicts (14) by comparing the third(from the bottom) socle layers of U4 and X since soc3(U4) sim= kG and soc3(X) sim= T (q)(since U4 is a submodule of X soc3(U4) rarr soc3(X) by [Lan83 ChapI Lemma 85(i)])This yields that such a U4 does not exist as a submodule of X Hence by (13)

dimk[EndkG(X)] = 3

However by (7) and Scottrsquos theorem on lifting of endomorphisms of p-permutation mod-ules [Lan83 Theorem II 124(iii)] this dimension has to be 4 so that we have a contra-diction As a consequence the second case in (9) does not occur This implies that onlythe first case in (9) can occur The claim follows

Proposition 54 Let G = PGL2(q) and Gprime = PGL2(qprime) where q and qprime are powers

of odd primes such that q equiv qprime (mod 4) and |G|2 = |Gprime|2 Let P be a common Sylow 2-subgroup of G and Gprime Then the Scott module Sc(GtimesGprime ∆P ) induces a splendid Moritaequivalence between B0(kG) and B0(kG

prime)

Proof Set B = B0(kG) Bprime = B0(kGprime) and M = Sc(G times Gprime ∆P ) By Proposi-

tion 47(c) M induces a stable equivalence of Morita type between B and Bprime Again weclaim that this stable equivalence is a Morita equivalence Let Q be a Sylow 2-subgroup ofB(q) Then it follows from Lemma 53 that the Loewy structures of H = H(PQ(kG)) =ΩQ(kG)kG and Hprime = H(PQ(kGprime)) = ΩQ(kGprime)kGprime are given by

H =kG oplus T (q)T (q)oplus kG

and Hprime =kGprime oplus T (qprime)T (qprime)oplus kGprime

Then it follows from Lemma 34(d) that


otimesB M =kGprime oplus T (qprime)T (qprime)oplus kGprime

oplus (proj)

Thus by the stripping-off method [KMN11 Lemma A1] and Lemma 34(a) we obtainthat

(T (q)oplus T (q))otimesB M = T (qprime)oplus T (qprime)oplus (proj)

Since T (q) is non-projective Theorem 21(a) implies that

(T (q)oplus T (q))otimesB M = T (qprime)oplus T (qprime)


Hence T (q)otimesB M = T (qprime) In addition kG otimesB M = kGprime by Lemma 34(a) Finally sinceIBr(B) = kG T (q) and IBr(Bprime) = kGprime T (qprime) it follows from Theorem 21(b) that Minduces a Morita equivalence between B and Bprime

6 The principal blocks of PSL2(q)⋊ Cf and PGL2(q)⋊ Cf

Let q = rm where r is a fixed odd prime number and m is a positive integer Wenow let H be one of the groups PSL2(q) or PGL2(q) and we assume moreover that aSylow 2-subgroup P of H is dihedral of order at least 8 We let G = H ⋊ Cf whereCf le Gal(FqFr) is as described in cases (D3)(i)-(ii) of Section 24 By the Frattiniargument we have G = NG(P )H therefore GH = NG(P )HH and we may assumethat we have chosen notation such that the cyclic subgroup Cf normalises P

Lemma 61 With the notation above we have G = H CG(P )

Proof The normaliser of P in G has the form

NG(P ) = P CG(P ) = P timesO2prime(CG(P ))

because Aut(P ) is a 2-group and P a Sylow 2-subgroup of G (see eg [Gor68 Lemma 541(i)-(ii)]) Therefore by the above the subgroup Cf le G centralises P so that we musthave

G = HCf le H CG(P )

and hence equality holds

We can now apply the result of Alperin and Dade (Theorem 22) in order to obtainsplendid Morita equivalences

Corollary 62 The principal 2-blocks B0(kG) and B0(kH) are splendidly Morita equiv-alent

Proof As G = H CG(P ) by Lemma 61 the claim follows directly from Theorem 22

7 Proof of Theorem 11

Proof of Theorem 11 First because we consider principal blocks only we may assumethat O2prime(G) = 1 Therefore we may assume that G is one of the groups listed in (D1)-(D3) in sect24 Now it is known by the work of Erdmann [Erd90] that the principal blocksin (1)-(6) fall into distinct Morita equivalence classes Therefore the claim follows directlyfrom Corollary 62 and Propositions 52 and 54

8 Generalised 2-decomposition numbers

Brauer in [Bra66 sectVII] computes character values at 2-elements for principal blockswith dihedral defect groups up to signs δ1 δ2 δ3 thus providing us with the generaliseddecomposition matrices of such blocks up to the signs δ1 δ2 δ3 As a corollary to Theo-rem 11 we can now specify these signs See also [Mur09 sect6] for partial results in thisdirection

Throughout this section we assume that G is a finite group with a dihedral Sylow2-subgroup P = D2n of order 2n ge 8 for which we use the presentation

P = 〈s t | s2nminus1



Furthermore we let ζ denote a primitive 2nminus1-th root of unity in C and we let z = s2nminus2

(see sect25)For a 2-block B of G we let Dgen(B) isin Matk(B)timesk(B) denote its generalised 2-decom-

position matrix In other words let S2(G) denote a set of representatives of the G-conjugacy classes of the 2-elements in a fixed defect group of B Let u isin S2(G) H =CG(u) and consider χ isin Irr(G) Then the generalised 2-decomposition numbers aredefined to be the uniquely determined algebraic integers duχϕ such that

χ(uv) =sum

ϕisinIBr(H)

duχϕϕ(v) for v isin H2prime

and we set Du = (duχϕ) χisinIrr(B)ϕisinIBr(bu)

where bu is a 2-block of H such that bGu = B so that

Dgen(B) =(duχϕ | χ isin Irr(B) u isin S2(G) ϕ isin IBr(bu)

)

is the generalised 2-decomposition matrix of B We recall that D1 is simply the 2-decomposition matrix of B By convention we see Dgen(B) as a matrix in Matk(B)timesk(B)

via Dgen(B) = (Du|u isin S2(G)) (one-row block matrix) Brauer [Bra66 Theorem (7B)]proved that B0(kG) satisfies

|Irr(B0(kG))| = 2nminus2 + 3

and possesses exactly 4 characters of height zero (χ0 = 1G χ1 χ2 χ3 in Brauerrsquos no-tation [Bra66]) In the sequel we always label the 4 first rows of Dgen(B0) with theseThe remaining characters are of height 1 and all have the same degree unless otherwisespecified we denote them by χ(j) with 1 le i le 2nminus2minus1 possibly indexed by their degreesThe first column of Dgen(B0(kG)) is always labelled with the trivial Brauer character

Corollary 81 The principal 2-block B0 = B0(kG) of a finite group G with a dihedralSylow 2-subgroup P = D2n of order 2n ge 8 affords one of the following generaliseddecomposition matrices

(a) If B0 simSM B0(kD2n) then

Dgen(B0) =

ϕ11 z = s2nminus2

sa t st1G 1 1 1 1 1χ1 1 1 1 minus1 minus1χ2 1 1 (minus1)a 1 minus1χ3 1 1 (minus1)a minus1 1χ(j) 2 2(minus1)j ζja + ζminusja 0 0

where 1 le j a le 2nminus2 minus 1 (up to relabelling the χirsquos)

(b) If n = 3 and B0 simSM B0(kA7) then

Dgen(B0) =

ϕ11 ϕ141 ϕ201 z = s2 s1G 1 0 0 1 1χ7 1 1 0 minus1 minus1χ8 1 0 1 1 minus1χ9 1 1 1 minus1 1χ5 0 1 0 2 0

where the irreducible characters and Brauer characters are labelled according tothe ATLAS [CCN+85] and the Modular Atlas [WPT+11] respectively


(c) If B0 simSM B0(k[PSL2(q)]) where (q minus 1)2 = 2n then

Dgen(B0) =

ϕ1 ϕ2 ϕ3 z = s2nminus2

sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1

χ(j)q+1 2 1 1 2(minus1)j ζja + ζminusja

where 1 le j a le 2nminus2 minus 1 χ1 and χ2 are labelled by their degrees and χSt is theSteinberg character

(d) If B0 simSM B0(k[PSL2(q)]) with (q + 1)2 = 2n then

Dgen(B0) =


sa

1G 1 0 0 1 1

χ(1)(qminus1)2 0 1 0 minus1 (minus1)a+1


χSt 1 1 1 minus1 minus1

χ(j)qminus1 0 1 1 2(minus1)j+1 (minus1)(ζja + ζminusja)


(e) If B0 simSM B0(k[PGL2(q)]) with 2(q minus 1)2 = 2n then

Dgen(B0) =

ϕ1 ϕ2 t z = s2nminus2

sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a

χ3 1 0 minus1 1 (minus1)a


where 1 le j a le 2nminus2 minus 1 χ1 and χ2 are labelled by their degrees and χ3 is alinear character

(f) If B0 simSM B0(k[PGL2(q)]) with 2(q + 1)2 = 2n then

Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 1 minus1 minus1

χ2 = χ(2)q 1 1 minus1 minus1 (minus1)a+1


χ(j)qminus1 0 1 0 (minus2)(minus1)j (minus1)(ζja + ζminusja)


Proof Generalised decomposition numbers are determined by a source algebra of theblock (see eg [The95 (4310) Proposition]) hence they are preserved under splendidMorita equivalences Thus by Theorem 11 for a fixed defect group P sim= D2n (n ge 3)if n = 3 there are exactly six respectively five if n ge 4 generalised 2-decompositionmatrices corresponding to cases (a) to (f) in Theorem 11


Let u isin S2(G) be a 2-element First if u = 1G then by definition Du = D1 is the 2-decomposition matrix of B0 Therefore in all cases the necessary information about D1 isgiven either by Erdmannrsquos work see [Erd77 TABLES] or the Modular Atlas [WPT+11]or [Bon11 Table 91] It remains to determine the matrices Du for u 6= 1 As we con-sider principal blocks only for each 2-element u isin P the principal block b of CG(u) isthe unique block of CG(u) with bG = B0 by Brauerrsquos 3rd Main Theorem [NT88 Theo-rem 561] Moreover when P = D2n then centralisers of non-trivial 2-elements alwayspossess a normal 2-complement so that their principal block is a nilpotent block [Bra64Corollary 3] It follows that duχ 1H

= χ(u)These character values are given up to signs δ1 δ2 δ3 by [Bra66 Theorem (7B) Theorem

(7C) Theorem (7I)] Thus we can use the character tables of the groups D2n A7 PSL2(q)and PGL2(q) (q odd) respectively to determine the signs δ1 δ2 δ3

(a) We may assume G = D2n Since G is a 2-group the generalised 2-decompositionmatrix Dgen(B0) is the character table of G in this case The claim follows

(b) We may assume G = A7 In this case D = D8 Using the ATLAS [CCN+85] wehave that the character table of B0 at 2-elements is

1a 2a 4a1G 1 1 1χ7 15 minus1 minus1χ8 21 1 minus1χ9 35 minus1 1χ5 14 2 0

Hence Dgen(B0) follows using eg the Modular Atlas [WPT+11] to identify therows of the matrix

(c) We may assume G = PSL2(q) with (q minus 1)2 = 2n The height 0 characters in B0

are 1G the Steinberg character χSt (of degree q) and the two characters of G ofdegree (q+1)2 Because the Steinberg character takes constant value 1 on sr (1 lea le 2nminus2) we have δ1 = 1 in [Bra66 Theorem (7B)] so that χ(j)(sa) = ζja + ζminusja

for every 1 le a le 2nminus2 And by [Bra66 Theorem (7C)] δ2 = δ3 = minus1 from which

it follows that χ(1)(q+1)2(s

a) = χ(1)(q+1)2(s

a) = (minus1)a for each 1 le r le 2nminus2 The

claim follows

(d) We may assume G = PSL2(q) with (q + 1)2 = 2n The height 0 charactersin B0 are 1G the Steinberg character χSt (of degree q) and the two charactersof G of degree (q minus 1)2 Because the Steinberg character takes constant valueminus1 on sa (1 le a le 2dminus2) we have δ1 = minus1 in [Bra66 Theorem (7B)] so thatχ(j)(sa) = (minus1)(ζja+ζminusja) for each 1 le a le 2dminus2 Then by [Bra66 Theorem (7C)]

δ2 = δ3 = 1 from which it follows that χ(1)(qminus1)2(s

a) = χ(1)(qminus1)2(s

a) = (minus1)a+1 for

every 1 le a le 2dminus2 The claim follows

(e) We may assume G = PGL2(q) with 2(q minus 1)2 = 2n The height one irreduciblecharacters in B0 have degree q + 1 The four height zero irreducible characters inB0 are the two linear characters 1G and χ3 (in Brauerrsquos notation [Bra66 Theorem

(7I)]) and the two characters of degree q say χ(1)q and χ

(2)q

Apart from 1G χ(1)q is the unique of these taking constant value 1 on sa (1 le a le

2dminus2) Therefore using [Bra66 Theorem (7B) Theorem (7C) Theorem (7I)] weobtain δ1 = 1 δ2 = minus1 = δ3 The claim follows


(f) We may assume G = PGL2(q) with 2(q + 1)2 = 2n The height one irreduciblecharacters in B0 have degree q minus 1 The four height zero irreducible characters inB0 are the two linear characters 1G and χ3 (in Brauerrsquos notation [Bra66 Theorem


(2)q Apart from 1G χ

(1)q is

the unique of these taking constant value 1 on sa (1 le a le 2nminus2) Therefore using[Bra66 Theorems (7B) (7C) and (7I)] we obtain δ1 = minus1 δ2 = 1 and δ3 = minus1The claim follows

For the character values of PSL2(q) we refer to [Bon11] and for the character values ofPGL2(q) we refer to [Ste51]

Acknowledgments

The authors thank Richard Lyons and Ronald Solomon for the proof of Lemma 42 The authorsare also grateful to Naoko Kunugi and Tetsuro Okuyama for their useful pieces of advice andto Gunter Malle for his careful reading of a preliminary version of this work Part of thiswork was carried out while the first author was visiting the TU Kaiserslautern in May andAugust 2017 who thanks the Department of Mathematics of the TU Kaiserslautern for thehospitality The second author gratefully acknowledges financial support by the funding bodyTU Nachwuchsring of the TU Kaiserslautern for the year 2016 when this work started Partof this work was carried out during the workshop rdquoNew Perspective in Representation Theoryof Finite Groupsrdquo in October 2017 at the Banff International Research Station (BIRS) Theauthors thank the organisers of the workshop Finally the authors would like to thank thereferees for their careful reading and useful comments

References

[AKO11] M Aschebacher R Kessar B Oliver Fusion Systems in Algebra and Topology LondonMath Soc Lecture Note Series vol391 2011

[AC86] M Auslander J Carlson Almost-split sequences and group rings J Algebra 103 (1986)122ndash140

[Alp76] JL Alperin Isomorphic blocks J Algebra 43 (1976) 694ndash698[Bon11] C Bonnafe Representations of SL2(Fq) Springer Berlin 2011[Bra64] R Brauer Some applications of the theory of blocks of characters of finite groups I J Algebra

1 (1964) 152ndash157[Bra66] R Brauer Some applications of the theory of blocks of characters of finite groups III J Al-

gebra 3 (1966) 225ndash255[Bra74] R Brauer On 2-blocks with dihedral defect groups in rdquoSymposia Mathrdquo Vol13 pp366ndash394

Academic Press New York 1974[BLO03] C Broto R Levi B Oliver The homotopy theory of fusion systems J Amer Math Soc

16 (2003) 779ndash856[Bro85] M Broue On Scott modules and p-permutation modules Proc Amer Math Soc 93 (1985)

401ndash408[Bro94] M Broue Equivalences of blocks of group algebras in rdquoFinite-dimensional Algebras and Re-

lated Topics (Ottawa 1992)rdquo Eds V Dlab and LL Scott Kluwer Academic Pub Dordrecht1994 pp1ndash26

[BP80] M Broue L Puig Characters and local structure in G-algebras J Algebra 63 (1980) 306ndash317

[CCN+85] JH Conway RT Curtis SP Norton RA Parker RA Wilson Atlas of Finite GroupsClarendon Press Oxford 1985

[CEKL11] DA Craven CW Eaton R Kessar M Linckelmann The structure of blocks with a Klein

four defect group Math Z 268 (2011) 441ndash476


[CG12] DA Craven A Glesser Fusion systems on small p-groups Trans Amer Math Soc 364(2012) no 11 5945ndash5967

[Dad77] EC Dade Remarks on isomorphic blocks J Algebra 45 (1977) 254ndash258[Erd77] K Erdmann Principal blocks of groups with dihedral Sylow 2-subgroups Comm Algebra 5

(1977) 665ndash694[Erd90] K Erdmann Blocks of tame representation type and related algebras Lecture Notes in Math-

ematics 1428 Springer-Verlag Berlin 1990[Gor68] D Gorenstein Finite Groups Harper and Row New York 1968[GW62] D Gorenstein JH Walter On finite groups with dihedral Sylow 2-subgroups Illinois J Math

6 (1962) 553ndash593[GW65] D Gorenstein JH Walter The characterization of finite groups with dihedral Sylow 2-

subgroups I II III J Algebra 2 (1965) 85ndash151 218ndash270 354ndash393[HK00] G Hiss R Kessar Scopes reduction and Morita equivalence classes of blocks in finite classical

groups J Algebra 230 (2000) 378ndash423[HK05] G Hiss R Kessar Scopes reduction and Morita equivalence classes of blocks in finite classical

groups II J Algebra 283 (2005) 522ndash563[IK17] H Ishioka N Kunugi Brauer indecomposability of Scott modules J Algebra 470 (2017)

441ndash449[Kes96] R Kessar Blocks and source algebras for the double covers of the symmetric and alternating

groups J Algebra 186 (1996) 872ndash933[Kes00] R Kessar Equivalences for blocks of the Weyl groups Proc Amer Math Soc 128 (2000)

337ndash346[Kes01] R Kessar Source algebra equivalences for blocks of finite general linear groups over a fixed

field Manusc Math 104 (2001) 145ndash162[Kes02] R Kessar Scopes reduction for blocks of finite alternating groups Quart J Math 53 (2002)

443ndash454[KKL15] R Kessar S Koshitani M Linckelmann On the Brauer indecomposability of Scott modules

Quart J Math 66 (2015) 895ndash903[KKM11] R Kessar N Kunugi N Mitsuhashi On saturated fusion systems and Brauer indecompos-

ability of Scott modules J Algebra 340 (2011) 90ndash103[Kos82] S Koshitani A remark on blocks with dihedral defect groups in solvable groups Math Z 179

(1982) 401ndash406[KK02] S Koshitani N Kunugi Brouersquos conjecture holds for principal 3-blocks with elementary

abelian defect group of order 9 J Algebra 248 (2002) 575ndash604[KMN11] S Koshitani J Muller F Noeske Brouersquos abelian defect group conjecture holds for the

sporadic simple Conway group Co3 J Algebra 348 (2011) 354ndash380[Lan83] P Landrock Finite Group Algebras and their Modules London Math Soc Lecture Note

Series 84 Cambridge University Press Cambridge 1983[Lin94] M Linckelmann A derived equivalence for blocks with dihedral defect groups J Algebra 164

(1994) 244ndash255[Lin96] M Linckelmann Stable equivalences of Morita type for self-injective algebras and p-groups

Math Z 223 (1996) 87ndash100[Lin96b] M Linckelmann The isomorphism problem for cyclic blocks and their source algebras In-

vent Math 12 (1996) 265ndash283[Lin01] M Linckelmann On splendid derived and stable equivalences between blocks of finite groups

J Algebra 242 (2001) 819ndash843[Lin18] M Linckelmann The Block Theory of Finite Group Algebras Cambridge Univ Press Cam-

bridge 2018 in press[Mur09] J Murray Real subpairs and Frobenius-Schur indicators of characters in 2-blocks J Algebra

322 (2009) 489ndash513[NT88] H Nagao Y Tsushima Representations of Finite Groups Academic Press New York 1988[Ple83] W Plesken Group Rings of Finite Groups Over p-adic Integers Lecture Notes in Math

1026 Springer-Verlag Berlin 1983[Pui82] L Puig Une conjecture de finitude sur les blocs unpublished manuscript 1982


[Pui94] L Puig On Joanna Scopesrsquo criterion of equivalence for blocks of symmetric groups AlgebraColloq 1 (1994) 25ndash55

[Pui99] L Puig On the Local Structure of Morita and Rickard Equivalences between Brauer BlocksBirkhauser Basel 1999

[Ric96] J Rickard Splendid equivalences Derived categories and permutation modules Proc LondonMath Soc 72 (1996) 331ndash358

[Rou01] R Rouquier Block theory via stable and Rickard equivalences in rdquoModular RepresentationTheory of Finite Groupsrdquo edited by MJ Collins et al Walter de Gruyter Berlin 2001pp101ndash146

[Ste51] R Steinberg The representations of GL(3 q) GL(4 q) PGL(3 q) and PGL(4 q) Canad JMath 3 (1951) 225ndash235

[Suz86] M Suzuki Group Theory II Springer Heidelberg 1986[The85] J Thevenaz Relative projective covers and almost split sequences Comm Algebra 13 (1985)

1535ndash1554[The95] J Thevenaz G-Algebras and Modular Representation Theory Clarendon Press Oxford 1995

[Tuv14] I Tuvay On Brauer indecomposability of Scott modules of Park-type groups J Group Theory17 (2014) 1071ndash1079

[Web82] P J Webb The Auslander-Reiten quiver of a finite group Math Z 179 (1982) 97ndash121[WPT+11] R Wilson J Thackray R Parker F Noeske J Muller F Lubeck C Jansen G Hiss

T Breuer The Modular Atlas Project httpwwwmathrwth-aachendesimMOC Version08042011

Shigeo Koshitani Center for Frontier Science Chiba University 1-33 Yayoi-cho

Inage-ku Chiba 263-8522 Japan

E-mail address koshitanmathschiba-uacjp

Caroline Lassueur FB Mathematik TU Kaiserslautern Postfach 3049 67653 Kaisers-

lautern Germany

E-mail address lassueurmathematikuni-klde

1 Introduction


21 Notation

22 Equivalences of block algebras

23 The Brauer construction and Brauer indecomposability

24 Principal blocks with dihedral defect groups

25 Dihedral 2-groups and 2-fusion



5 The principal blocks of `39`42`613A``45`47`603APSL2(q) and `39`42`613A``45`47`603APGL2(q)

6 The principal blocks of `39`42`613A``45`47`603APSL2(q)Cf and `39`42`613A``45`47`603APGL2(q)Cf



Acknowledgments

References


In this paper we investigate the principal blocks of groups G with a Sylow 2-subgroup Pwhich is dihedral of order at least 8 over an algebraically closed field k of characteristic 2Our main result is the following

Theorem 11 Let n ge 3 be a positive integer Then the splendid Morita equivalenceclasses of principal 2-blocks of finite groups with dihedral defect group D2n of order 2n

coincide with the Morita equivalence classes of such blocks More accurately a principalblock with dihedral defect group D2n is splendidly Morita equivalent to precisely one of thefollowing blocks

(1) kD2n(2) B0(kA7) in case n = 3(3) B0(k[PSL2(q)]) where q is a fixed odd prime power such that (q minus 1)2 = 2n(4) B0(k[PSL2(q)]) where q is a fixed odd prime power such that (q + 1)2 = 2n(5) B0(k[PGL2(q)]) where q is a fixed odd prime power such that 2(q minus 1)2 = 2n or(6) B0(k[PGL2(q)]) where q is a fixed odd prime power such that 2(q + 1)2 = 2n

In particular if q and qprime are two odd prime powers as in (3)-(6) such that either(q minus 1)2 = (qprime minus 1)2 (Cases (3) and (5)) or (q + 1)2 = (qprime + 1)2 (Cases (4) and (6)) thenB0(k[PSL2(q)]) is splendidly Morita equivalent to B0(k[PSL2(q

prime)]) and B0(k[PGL2(q)]) issplendidly Morita equivalent to B0(k[PGL2(q

prime)])

Remark 12 We note that if G is a soluble group and B is an arbitrary 2-block of Gwith a defect group P sim= D2n with n ge 3 which is not nilpotent then n = 3 and B isactually splendidly Morita equivalent to kS4

sim= k[PGL2(3)] (see [Kos82]) There is alsoan interesting and related result by Linckelmann [Lin94] where all derived equivalenceclasses of blocks B with dihedral defect groups over the field k are determined

Furthermore we will prove in Corollary 47 that for a given defect group P sim= D2n

(n ge 3) up to stable equivalence of Morita type there are exactly three equivalenceclasses of principal blocks of finite groups G with defect group P and these depend onlyon the fusion system FP (G) or equivalently on the number of modular simple modulesin B0(kG)

In order to prove Theorem 11 we will construct explicit Morita equivalences inducedby bimodules given by Scott modules of the form Sc(GtimesGprime∆P ) First we will constructstable equivalences of Morita type using these modules using gluing methods and thendetermine when these stable equivalences are actually Morita equivalences To reach thisaim we make use of the notion of Brauer indecomposability introduced in [KKM11] Inparticular we will use some recent results of Ishioka and Kunugi [IK17] in order to provethe following theorem

Theorem 13 Let G be a finite group with a dihedral 2-subgroup P of order at least 8Assume moreover that the fusion system FP (G) is saturated and CG(Q) is 2-nilpotentfor every FP (G)-fully normalised non-trivial subgroup Q of P Then the Scott moduleSc(G P ) is Brauer indecomposable

This result crucial for our work may in fact be of independent interest as it is anextension of the main results of [KKL15] We note that further results on Brauer in-decomposability of Scott modules under different hypotheses may be found in [KKM11Theorem 12] [KKL15 Theorem 12(b)] and [Tuv14]














and [The95]





















V (Q) = V Q sum

RltQ

TrQR(VR)











group Q of P










number

























s2nminus2


where s2nminus2





prime)


have







)(∆〈z〉)








M(∆〈z〉)∣∣∣M(∆〈z〉)











= kGdarrQuarrGprime

= kQuarrGprime









0 otherwise

Namely







kGprime)

where


uarrGprime

= kQiuarrG

prime















= UdarrPuarrGprime


=oplus

gisin[QGP ]

kQgcapPuarrGprime







= dim[HomB(UQ kG)]

= dim[HomBprime

(F (UQ) F (kG)

)]

= dim[HomBprime

(F (UQ) kGprime

)]

= dim[HomBprime

(F (UQ) kGprime

)]








Gprime























to say we have


RM(∆R)lowast

Moreover

(2) BR(∆Q) = BQ



RM(∆R)lowast




(4) (M(∆R)) (∆Q) = M(∆Q)

Hence

BQ = (BR)(∆Q) by (2)


RM(∆R)lowast

)(∆Q) by (1)






Q











NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R



G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L










Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References














and [The95]





















V (Q) = V Q sum

RltQ

TrQR(VR)











group Q of P










number

























s2nminus2


where s2nminus2





prime)


have







)(∆〈z〉)








M(∆〈z〉)∣∣∣M(∆〈z〉)











= kGdarrQuarrGprime

= kQuarrGprime









0 otherwise

Namely







kGprime)

where


uarrGprime

= kQiuarrG

prime















= UdarrPuarrGprime


=oplus

gisin[QGP ]

kQgcapPuarrGprime







= dim[HomB(UQ kG)]

= dim[HomBprime

(F (UQ) F (kG)

)]

= dim[HomBprime

(F (UQ) kGprime

)]

= dim[HomBprime

(F (UQ) kGprime

)]








Gprime























to say we have


RM(∆R)lowast

Moreover

(2) BR(∆Q) = BQ



RM(∆R)lowast




(4) (M(∆R)) (∆Q) = M(∆Q)

Hence

BQ = (BR)(∆Q) by (2)


RM(∆R)lowast

)(∆Q) by (1)






Q











NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R



G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L










Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References










V (Q) = V Q sum

RltQ

TrQR(VR)











group Q of P










number

























s2nminus2


where s2nminus2





prime)


have







)(∆〈z〉)








M(∆〈z〉)∣∣∣M(∆〈z〉)











= kGdarrQuarrGprime

= kQuarrGprime









0 otherwise

Namely







kGprime)

where


uarrGprime

= kQiuarrG

prime















= UdarrPuarrGprime


=oplus

gisin[QGP ]

kQgcapPuarrGprime







= dim[HomB(UQ kG)]

= dim[HomBprime

(F (UQ) F (kG)

)]

= dim[HomBprime

(F (UQ) kGprime

)]

= dim[HomBprime

(F (UQ) kGprime

)]








Gprime























to say we have


RM(∆R)lowast

Moreover

(2) BR(∆Q) = BQ



RM(∆R)lowast




(4) (M(∆R)) (∆Q) = M(∆Q)

Hence

BQ = (BR)(∆Q) by (2)


RM(∆R)lowast

)(∆Q) by (1)






Q











NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R



G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L










Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References







number

























s2nminus2


where s2nminus2





prime)


have







)(∆〈z〉)








M(∆〈z〉)∣∣∣M(∆〈z〉)











= kGdarrQuarrGprime

= kQuarrGprime









0 otherwise

Namely







kGprime)

where


uarrGprime

= kQiuarrG

prime















= UdarrPuarrGprime


=oplus

gisin[QGP ]

kQgcapPuarrGprime







= dim[HomB(UQ kG)]

= dim[HomBprime

(F (UQ) F (kG)

)]

= dim[HomBprime

(F (UQ) kGprime

)]

= dim[HomBprime

(F (UQ) kGprime

)]








Gprime























to say we have


RM(∆R)lowast

Moreover

(2) BR(∆Q) = BQ



RM(∆R)lowast




(4) (M(∆R)) (∆Q) = M(∆Q)

Hence

BQ = (BR)(∆Q) by (2)


RM(∆R)lowast

)(∆Q) by (1)






Q











NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R



G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L










Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References













s2nminus2


where s2nminus2





prime)


have







)(∆〈z〉)








M(∆〈z〉)∣∣∣M(∆〈z〉)











= kGdarrQuarrGprime

= kQuarrGprime









0 otherwise

Namely







kGprime)

where


uarrGprime

= kQiuarrG

prime















= UdarrPuarrGprime


=oplus

gisin[QGP ]

kQgcapPuarrGprime







= dim[HomB(UQ kG)]

= dim[HomBprime

(F (UQ) F (kG)

)]

= dim[HomBprime

(F (UQ) kGprime

)]

= dim[HomBprime

(F (UQ) kGprime

)]








Gprime























to say we have


RM(∆R)lowast

Moreover

(2) BR(∆Q) = BQ



RM(∆R)lowast




(4) (M(∆R)) (∆Q) = M(∆Q)

Hence

BQ = (BR)(∆Q) by (2)


RM(∆R)lowast

)(∆Q) by (1)






Q











NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R



G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L










Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References





M(∆〈z〉)∣∣∣M(∆〈z〉)











= kGdarrQuarrGprime

= kQuarrGprime









0 otherwise

Namely







kGprime)

where


uarrGprime

= kQiuarrG

prime















= UdarrPuarrGprime


=oplus

gisin[QGP ]

kQgcapPuarrGprime







= dim[HomB(UQ kG)]

= dim[HomBprime

(F (UQ) F (kG)

)]

= dim[HomBprime

(F (UQ) kGprime

)]

= dim[HomBprime

(F (UQ) kGprime

)]








Gprime























to say we have


RM(∆R)lowast

Moreover

(2) BR(∆Q) = BQ



RM(∆R)lowast




(4) (M(∆R)) (∆Q) = M(∆Q)

Hence

BQ = (BR)(∆Q) by (2)


RM(∆R)lowast

)(∆Q) by (1)






Q











NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R



G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L










Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References




kGprime)

where


uarrGprime

= kQiuarrG

prime















= UdarrPuarrGprime


=oplus

gisin[QGP ]

kQgcapPuarrGprime







= dim[HomB(UQ kG)]

= dim[HomBprime

(F (UQ) F (kG)

)]

= dim[HomBprime

(F (UQ) kGprime

)]

= dim[HomBprime

(F (UQ) kGprime

)]








Gprime























to say we have


RM(∆R)lowast

Moreover

(2) BR(∆Q) = BQ



RM(∆R)lowast




(4) (M(∆R)) (∆Q) = M(∆Q)

Hence

BQ = (BR)(∆Q) by (2)


RM(∆R)lowast

)(∆Q) by (1)






Q











NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R



G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L










Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References




= dim[HomB(UQ kG)]

= dim[HomBprime

(F (UQ) F (kG)

)]

= dim[HomBprime

(F (UQ) kGprime

)]

= dim[HomBprime

(F (UQ) kGprime

)]








Gprime























to say we have


RM(∆R)lowast

Moreover

(2) BR(∆Q) = BQ



RM(∆R)lowast




(4) (M(∆R)) (∆Q) = M(∆Q)

Hence

BQ = (BR)(∆Q) by (2)


RM(∆R)lowast

)(∆Q) by (1)






Q











NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R



G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L










Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References











to say we have


RM(∆R)lowast

Moreover

(2) BR(∆Q) = BQ



RM(∆R)lowast




(4) (M(∆R)) (∆Q) = M(∆Q)

Hence

BQ = (BR)(∆Q) by (2)


RM(∆R)lowast

)(∆Q) by (1)






Q











NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R



G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L










Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References








NG(Q)

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

K ⋊NP (Q) = K ⋊D8

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

CG(Q) = K ⋊ R



G = NG(Q)L

S3

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

R = (K ⋊NP (Q))L

C2

⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧

Q = CG(Q)L










Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References








Q le P




























prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References



















prime)











M(∆〈z〉) = Mz





prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References




prime




Moreover we have


prime)


prime

) (∆P )(ggprime))












M(∆〈t〉) = Mt



B and Bprime






prime)




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References




prime)







that


prime))











Sc(G B(q)) =kG


kG












kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References










kG

kGS(q)S(q)lowast

kG

kGS(q)lowast

S(q)kG



kG oplus S(q)





Y =S(q)kG

oplus S(q)lowast


Z =S(q)kG


W =kG

S(q)lowastoplus kG


U =kG

S(q)lowast



Sc(G B(q)) =kG


kG

as desired




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References




prime)











kGprime

Thus by Lemma 34(d)


kGotimesB M =


kGprime

oplus (proj)








Sc(G B(q)) =

kG


kG







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References







X = kG + kG + (kG + T (q)) + (kG + T (q))


Y =kH


kH


P (kG) = Sc(GB)


Sc(G B) = P (kG) =

kG

kGT (q)

oplusT (q)kG

kG


XX1sim=

kGkG


sim=kGkG


dim[HomkG(XkGkG

)] = 2


kGkG



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References



X =


kG




(8) XX2 =kG

kG oplus T (q)T (q)


(9)


kG

or

kGkG oplus T (q)

T (q)kGkG



(10) X3 =T (q)

kG oplus T (q)kG

(socle series)


(11) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


X =

kG

kGT (q)kG

oplus T (q)

kG

=

kGkG oplus T (q)

T (q)kGkG

(Loewy series) =

kGkGT (q)

kG oplus T (q)kG

(socle series)



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References



(13) U1 = X U2sim= kG U3

sim=kGkG

U4sim=

kGkGT (q)kG


(14) X =

kGkGT (q)

kG oplus T (q)kG

(socle series)


dimk[EndkG(X)] = 3




prime)








oplus (proj)













G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References









G = HCf le H CG(P )
















χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References





χ(uv) =sum

ϕisinIBr(H)





)







Dgen(B0) =

ϕ11 z = s2nminus2




Dgen(B0) =





Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References



Dgen(B0) =


sa

1G 1 0 0 1 1

χ1 = χ(1)(q+1)2 1 1 0 1 (minus1)a

χ2 = χ(2)(q+1)2 1 0 1 1 (minus1)a

χ3 = χSt 1 1 1 1 1




Dgen(B0) =


sa

1G 1 0 0 1 1







Dgen(B0) =


sa

1G 1 0 1 1 1

χ1 = χ(1)q 1 1 minus1 1 1

χ2 = χ(2)q 1 1 1 1 (minus1)a





Dgen(B0) =


sa

1G 1 0 1 1 1



















a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References













a) = χ(1)(q+1)2(s


claim follows








(2)q







(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References





(1)q is



Acknowledgments


References
























































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References











































lautern Germany


1 Introduction


21 Notation











Acknowledgments

References

with dihedral defect groups - arxiv.org · 2 s.koshitaniandc.lassueur in this paper, we investigate...

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