theorems like sylow's (hall)

8/18/2019 Theorems Like Sylow's (Hall)

1/19

THEOREMS LIKE SYLOW'S

By

P.

HALL

[Received 24 June 1955.—Read 17 November 1955]

1. Discussion

of

results

1.1. Notation.

Let

w

be any set of

primes

and let

w'

be the

comple-

mentary

set

consisting

of all

primes

not in

w. Ev ery positive integer

m

can be expressed uniquely in the form

m

=

m

m

m

VT

,,

where m

m

is the largest divisor of m which has no prime divisor in m'.

We denote the order of a finite group G by (G). If (6%, = (G ) we call

G a w-group. If

H

is a subgroup of G such tha t

(H) = (G)^,

we call

H

an

S

m

-subgroup of G. In

particular, when w consists

of a

single prime p,

an

$p-subgroup

is the

same

as a

Sylow ^-subgroup.

But for

general

-nr, a

Sylow m-subgroup

is

not the same as an ^ - sub grou p , bu t

is

an $p-subgroup

for some prime ^in-nr.

Let E

m)

C

m

, and

D

m

be the

following propositions about

a

finite group G.

E

m

: G has at least one ^ - s u b g r o u p .

C

m

: G satisfies E

m

and any two /S^-subgroups of G are conjugate in G.

D

m

: G satisfies C

m

and every -nr-subgroup of G is contained in some

#

OT

-subgroup

of

G.

Obviously D ̂can also be expressed in either of the alternative forms:

(i) G satisfies C

m

and

every maximal -nr-subgroup

of G is an

#

OT

-subgroup

of G;

(ii)

G satisfies E

m

and, if

H

is an

^ - s u b g r o u p

of

G

and

L

is any

xir-subgroup of G, then

L

^

H

x

for some

x e G.

1.2. D-theorems. A sufficient condition for a finite group to satisfy

D

w

m ay be called a D-theorem. Similarly for ^-theorems and C-theorems.

Our main object is to prove a new D-theorem, viz. Theorem D5, below.

The two basic D-theorems are, of course, those due to Sylow and Schur,

viz.

THEOREM

Dl. If p is a prime, then every finite group sa tisfies D

p

.

THEOREM

D2. / / G has a n ormal Abelian 8

m

.-subgroup, then G satisfies D

m

.

We mak e once

and for all the

obvious remark th at,

if

m

x

and

m

2

are two

sets

of

primes such that (G)

mi

— (G)^, then

for

this particular group G

the propositions D

mi

and J)

m%

are equivalent: G satisfies bo th or neither.

Similarly for the corresponding E-

and

C-propositions. So Sylow

3

s theorem

Proc. London Math. Soc. 3) 6 1956)


2/19

THE OR EM S LI K E SYLO W'S 287

may be expressed by saying that G satisfies D

m

whenever at most one

prime in

m

divides

(G).

All -nr-groups satisfy D

m

trivially.

In an earlier paper (9), we proved

THEOREM

D 3.

If G is soluble, then G satisfies D

m

for all

m.

Quite recently, W ielandt (14) has derived a remark able new D -theorem,

which resembles Sylow's theorem and differs from D2 and D3 in the fact

that it does not reduce to a triviality for simple groups.

We define the propositions E%,

E^, 0%,

and

D

s

m

abo ut a finite grou p

G

as follows.

E^\ G has a ni lpotent ^- su bg ro up .

E% f\ G has a soluble ^- su bg ro up .

C

s

m

: G satisfies C

m

and its /S^-subgroups are soluble.

D ^ : G satisfies D^ and its -nr-subgroups are soluble.

Wielandt's theorem is

THEOREM D4. E% implies D

m

.

This makes it superfluous to introduce the propositions C^. and D

7

^,

analogous to

C

s

m

and

D%

but with 'soluble' replaced by 'nilpotent', since

they would both be equivalent to

E^.

1.3. That E% . does not in general imply D

m

or even C

m

is shown, as h as

been remarked before, by the simple group of order 168 which has two

distinct classes of conjugate o ctahed ral subgroups. This grou p satisfies

Uf.3 but not O

2

,3-

Wielandt suggests as a possibility that the presence of a supersoluble

^-subgroup might be sufficient to imply

C

m

or perhaps even

D

m

.

B u t

this is not the case, as is shown by the following examples.

We denote by

L

p

the simple group of order

%p(p

2

—l),

where

p

is a prim e

greater tha n 3. Then L

lly

of order 660, has two distinct classes of con jugate

$

2 3

-subgroups, one class being dihedral (and therefore supersoluble) and

the other class tetrah edra l. Again, L

ei

has two distinct classes of conjugate

$

2 3 5

-subgroups, one class being dihedral and the other icosahedral (and

therefore not even soluble). For these properties of L

p

, see Burnside (1),

Chapter XX.

However, it is easy to see that any two supersoluble S

m

-subgroups must

be conjugate. A more general result m ay be described as follows. Le t

p

x

, p

2

,..., p

r

be distinct primes. We say th at a finite group H has a Sylow

series of complexion (p

lt

p

2

,---, p

r

) if (H ) is divisible by no primes other

than p

1}

p

2

, -, p

r

and if, for each i = 1,

2,...,

r—1 H has a normal (and

therefore characteristic)

8

PltPt

^-subgroup. Then we may state :


3/19

288 P. H A L L

T H E O R E M

A l .

Let p

v

p^,..., P

r

be the distinct primes which divide (G)

m

,

arranged in any given order. Then any two S

m

-subgroups of G both of w hich

have a Sylow-series of complexion (p

v

p

2

,--, p

r

) must be conjugate in 0.

The conjugacy of supersoluble /S^-subgroups follows from this theorem,

because a supersoluble group of order pi

1

p%

i

...p%

r

, where p

x

> p

2

> ... > p

r

,

always has a Sylow series of complexion (Pi,P2,---,p

r

)'

The group

L

n

does no t con tradict Theorem

A1

because its dihedral

#

2

g-subgroups h ave Sylow series of complexion (3,2) bu t none of com-

plexion (2,3), whereas its tetrahedral $

2)3

-subgroups have Sylow series of

complexion (2,3) but none of complexion (3,2).

I know of no counter-example to the conjecture that, if 2 does not

belong to m, then E

m

implies D

m

.

1.4. The principal theorem of the present pape r is

THEOREM D5 . / /

K is a normal subgroup of 0 such that K satisfies Ei^

and G/K satisfies D^, then G satisfies D% .

Here we merely note a few corollaries of this theorem.

A chain of subgroups

G = G

o

>

G

t

> ... >

G

r

=

1 of a group

G

will

be called a series of G if each term G

i

is norm al in the p receding te rm G^

v

thoug h n ot necessarily norm al in

G.

C OR OL L AR Y

D 5 . 1 . / /

G has a series G = G

Q

>

G

x

> ... >

G

r

=

1

such

that each of the factor groups

G

i

_

x

jG

i

satisfies E

1

^, then G satisfies D^.

It would be sufficient here to postulate that the first factor group of the

series,

G/G

1}

satisfies the weaker condition

D%

instead of

E%.

Following Cunihin (2), we say that a finite group is vr-separable if all

its c.f. (= composition factors) are Ttr'p-groups for various primes

p

in

m.

Thus G is tn--separable if and only if no two distinct primes in m divide

the order of any c.f. of G. If p is a prime, all finite grqmps are ^-separable

If G is both -nr-separable and xo-'-separable, then G is soluble: for th en the

order of any c.f. of G must have the form p

a

qP with p in TO- and qin m'\

but by a classical theorem of Burnside, groups of order p^qP are always

soluble, so th a t either a = 1, fi = 0 or else a = 0, £ = 1.

Clearly, if m

x

is any subset of m, then every m-separable group is also

•ro-j-separable. In (2), Theorem X I, Cunihin proves th a t all xtr-separable

groups satisfy C^, and therefore, by the preceding remark, also C

s

mx

for

any subset TD-J of m. This result is contained in

C OR OL L AR Y D 5 . 2 . All -m-separable groups satisfy

D

s

mi

,

for any subset

TO^

o

m


4/19

THEO REM S LIK E SYLOW 'S 289

This corollary also contains as a special case the main Z)-theorem of

Cunihin's paper, Theorem XII, (1) and (2).

We call a finite group

G m-serial

if every c.f. of

G

is either a -nr-group

or else a -ur'-group. Equivalently, G is -nr-serial if it has a m-series, i.e. a

series G = G

o

> G

x

> ... > G

r

= 1 such tha t each of the factor groups

G

i

_

1

IG

i

is eithe r a xtr-group or else a -nr'-group. Clearly, -nr-serial groups

are the same as m'-serial groups. If p is any prime in w and q any prime

in xtr', then every xtr-serial group is p, ^-separable. Hence we have

COROLLARY D5.3. All m-serial groups satisfy D ,

pq

for any p in m and

any a in m'.

We note that, by the theorem of Burnside already mentioned, D

p

q

is

equivalent to D

s

vq

.

1.5. In his well-known textbook (15), Chapter IV, Satz 27, Zassenhaus

proves two C-theorems. The corresponding D-theorem s, which follow a t

once,

are

THEOREM

D6. / / G has a normal

8^-subgroup K such that G/K is soluble,

then G satisfies D

m

.

THEOREM D7. If G has a normal

S

m

>-subgroup

K such that K is soluble,

then G satisfies D

m

.

We call a finite group G vx-soluble if it is both -nr-separable and xo--serial.

Equivalently, G is xtr-soluble if every c.f. of G is either a ^p-group (and

therefore cyclic of order p) for some p in vx, or else a xcr'-group.t

In (3), Cunihin proves the following generalizations of D6 and D7.

THEOREM D6.* Every m-soluble group satisfies D^.

1

for any subset m

x

o f -m .

THEOREM

D7.*

Every m-soluble group satisfies D^Jor any subset TJT

X

of m .

Of these resu lts, Theorem D6* is contained as a special case in Corollary

D5.2.

Fo r the sake of completeness we include in section 2.5 th e deduction

of Theo rem

D7*

from Sch ur's Theorem D 2. Both D6* and

D 7*

are generali-

zations of Theorem D 3.

1.6. The theory of systems of permutable Sylow subgroups, developed

in (10) for the case of soluble groups, extends at once in an appropriate

form to all groups satisfying C

s

m

. This has been rem arked for th e case of

m-separable groups by Gol'berg (8).

t This terminology seems preferable to that of Hall and Higman (12). In that

paper, nr-serial groups as defined above were called -nr-soluble. B ut th e m -subgroups

of m -serial groups need no t always be soluble, whereas all m-su bgroups of a itr-separ-

able group are soluble, and in particular, all m-subgroups of a m-soluble group are

soluble. Owing to Burnside's theorem, no distinction arises between -nr-serial and

-nr-soluble groups as denned here unless -nr conta ins at least three prim es. I hope

that the present choice of terms is reasonably consistent with Cunihin's.

5388.3.6 XJ


5/19

290

P . HA LL

A set of Sylow subgroups

P

i

of the finite group

G,

one for each p rime

p

i

in

w,

which are permutable in pairs (i.e. such that

P

t

Pj = P

i

P

i

for all

i, j),

will be called a Sylow m-system of G. Obviously, only a finite number of

the Pi can differ from 1. Owing to the pe rm utab ility of the P

i}

their pro duct

is an ^-su bg ro up of G and it is shown in (10), § 2, th at this product m ust

be soluble. Conversely, a soluble iS^-subgroup H of G has a Sylow m-system

(whose product is therefore H itself) and this will also be necessarily a

Sylow -nr-system of G as defined above.

In (10), § 4, it was shown that any two Sylow -nr-systems (...,

P

i}

...)

and

(...,

Qi,...)

of a soluble -nr-group

H

are conjugate in

H,

in the obvious sense

that there is an element

x

in

H,

independent of

i,

such that

Q

t

= P\

for

all i. Thus we may state at once:

T H E O R E M A2. G has at least one Sylow m-system if and on ly if it satisfies

E^. G has one and only one class of conjugate Sylow m-systems if and only

if it satisfies C^.

Theorem

D 3 allows the following conclusions to be draw n.

(i)

E% implies E^ for an y subs et m

1

of to-.

(ii)

D^ implies D*

ni

for any subset m

1

of TO-.

(i)

is immediate. As for (ii), let

G

satisfy

D

s

m

and let

L

be a maximal

xo-j-subgroup

of G. Then L is contained in. some /S^-subgroup H of G.

B ut

H is soluble. Hen ce, by Th eorem D 3, L is an

jS^-subgroup

of H and

therefore

also of

G.

S ince t he ^ - s u bg ro u ps of

H

are all conjugate in

H

and

t h e ^ - s u b g r o u p s H of G are all conjugate in G, it follows th a t G has

only

a single class of conjugate m axim al

TOi-subgroups

and th at these are

soluble iS^-subgroups of G. Thus & in fa ct satisfies Z )^ .

Cunihin

(2) defines a finite gro up to be

m-Sylow-regular

if it satisfies

C^

JTl

for

every subset

m

1

of

m.

Thus we conclude th at

all groups satisfying D^

are m -Sylow-regular; an d further, from Th eorem

A2,

all such groups possess

a single class of conjugate Sylow m

x

-systems for any given subset w

1

of

xtr.

In particular, this is the case for all groups G satisfying the hy pothesis of

Theorem D5 or Corollary D5.1. This statement is a generalization of

Gol'berg's result about -nr-separable groups. It is obvious that in a

itr-Sylow-regular grou p, the Sylow T&i-systems are all to be ob tained from

the Sylow xo--systems by deleting the appropriate terms.

It must be noted, however, that a group may very well satisfy C^ for

some suitable

-m

w itho ut being -nr-Sylow-regular in C unih in's sense. Fo r

example, the simple group

L

83

has a single class of conjugate subgroups

of order 84. These subgroups are dihedral, so that L

83

satisfies Ci^j. But

L

83

has two distinct classes of conjugate subgroups of order 12 and there-

fore does not satisfy C

23

. Thus

L

83

is not 2,3,7-Sylow-regular.


6/19


1.7. E-Theorems. Ev ery Z)-theorem includes a C-theorem and a n

jEJ-theorem by im plication. But , of course, an ^/-theorem can som etimes be

proved from hypotheses too weak to yield the corresponding D-result.

The basic ^-theorem is Schur's,

THEOREM E l . If G has a normal S

m

,-subgroup, then G satisfies E

m

.

This may be expressed in a more general form as follows:

THEOREM E l . * If K is a normal subgroup o f G such that G/K is a m-group,

then G has a m-subgroup L such that KL = G and K n L is nilpotent.

From this we may deduce the following result, suggested by Cunihin's

theorems on the factorization of -nr-separable groups (4).

THEOREM A3. Let m

0

,

in

lt

and vr

2

be mutually exclusive sets of primes.

If every c.f. of the inite group G is either a

tcr

0

m^group or else a

vr

0

m

2

-group,

then G

=

HK, where H is a -m

0

^-subgroup, K is a w

0

m

2

-subgroup,

and

H n K is a soluble m

0

-subgroup.

An immediate consequence of Theorem El is the well-known

THEOREM E 2 .

If K is a normal subgroup of G such that K satisfies G

m

and G/K satisfies E

m

, then G satisfies E

m

.

I t is no t sufficient in this theorem to assume th a t both

K

and

G/K

satisfy

E

m

. For example, the group of automorph isms L, of orde r 336, of the simple

group L = L

7

, of orde r 168, does not satisfy E

23

, although both L and L/L

do so; because the outer automorphisms of L interchange the two classes

of conjugate octahedral subgroups in L.

It will be convenient to mention here the following corollaries of

Theorem E2.

COROLLARY E2 .1 . / / all the c.f. of G satisfy C

m

, then G satisfies E

m

.

COROLLARY

E2.2. ' / /

the c.f. of G are either m-groups or else m'p-groups

for various primes p in w, then G satisfies E

m

.

COROLLARY E2 .3 . / / G is in-serial and m

x

= m, m', mq, TJJ'P, or pq,

where p is in w and q in -&•', then G satisfies E

mi

.

I t w as shown in (11) th at , if a finite grou p G satisfies E

p

. for all primes p,

then

G

is soluble. More generally, by a ve ry similar argum ent, it is easy

to see that, if G satisfies E

m

and also E

p

. for all primes p in -nr, then G is

m-separable.

It may well be conjectured that

G

is soluble whenever it

satisfies

E

pq

for all pairs of prim es

p

and

q

which divide its order. Some

light may perhaps be thrown on this question by studying the behaviour

of the better known insoluble groups in relation to the propositions E

p q

.

Here we only consider the symmetric groups and prove:

THEOREM A4. Let S

n

be the symmetric group of order

n\

and letp


7/19

292 P. H A L L

where p and q are primes. Then S

n

satisfies E

lhq

only ivhen p = 2, q = 3 ,

and n

= 3, 4, 5, 7,

and

8.

For n = 5,1, and

8, E

u

satisfies

C|,3

but not

D

2 3

.

Taken in conjunction with Theorem D3, this result shows that, if

m

involves more than two primes not exceeding

n,

an ^-subgroup of S

?l

,

supposing one to exist, can only be insoluble. An exam ple would be the

£y-subgroups of S

p

, where p is a prime greater than 5. These are iso-

morphic with ^

p

-

x

.

1.8. C-Theorems. In a long series of papers (see particularly 2 , 3 ,5 , 6 , 7 ) ,

Cunihin proves a num ber of interesting (7-theorems, of which we m ention

two,

deducible from Zassenhaus's Theorems D6 and D7, respectively.

Let C%, be the following proposition about a finite group G.

C^: G satisfies C

m

and NJT is soluble, where T i s any ^-subgroup of

G and N is its normalizer in G.

Then the theorems in question are as follows.

T H E O R E M

C l .

If K is a normal subgroup of G such that K satisfies C

m

and G/K satisfies C

s

m

, then G satisfies C

m

.

T H E O R E M C 2. If K is a normal subgroup of G such that K satisfies C^

and G/K satisfies C

m

, then G satisfies C

m

.

I t is a corollary of Theorem Cl t hat , if G has a series whose factor groups

all satisfy C

s

m

, then G itself satisfies C

s

m

. It is a corollary of Theorem C2

that, if G has a series whose factor groups all satisfy C

s

m

, then G itself

satisfies C^. In view of Corollary E2 .1, we can also say th at , if G has a

series whose factor groups all satisfy G

m

, and if all m-subgroups of G a re

soluble, then G will satisfy

C%\

in view of Theorem El, we can add that,

if

G

has a series whose factor groups all satisfy

C

m

and if all xu'-subgroups

of

G

are soluble, then

G

will satisfy C^..

It is natural to make the

C O N J E C T U R E

C 3 .

If K is a normal subgroup of G such that both K and

G/K satisfy C

m

, then G satisfies C^.

The argument which yields Theorems Cl and C2 from Theorems D6

and D7 shows that this conjecture is equivalent to the special case already

mentioned by Zassenhaus, viz.

C O N J E C T U R E C 3 . * / / G has a normal S

m

.-subgroup K, then G satisfies C

m

.

Zassenhaus states, in (15), p. 126, th at E . W itt has shown how to deduce

C3*

from the still more special case where K is a simple group of composite

order and the centralizer of

K

in G is 1. We give here a proof of this red uc-

tion in a somewhat more precise form.

Let us say that a group G involves an abstract group V if there is a


8/19

THEOREMS LIKE SYLOW'S 293

subgroup

H

of

G

and a normal subgroup

K

of

H

such that

H/K

~ F.

If in addition

(T) < (G), we

shall say that

G involves V properly.

We define a finite group G to be

m-exceptional

if it satisfies the following

three conditions:

(i)

G

has a normal ^/- subgroup ,

(ii)

G

does not satisfy

C

m

.

(iii) Every group F properly involved in G satisfies C

m

.

It is clear that, if

G

satisfies (i) and involves F, then F will also satisfy

(i).

Hence every group which satisfies both (i) and (ii) involves a zcr-excep-

tional group. Thus Conjectures C3 and C3* are equivalent to the state-

ment that xtr-exceptional groups do not exist. We prove

T H E O R E M A 5 .

Let

G be m-exceptional. Then both the normal S

m

.-subgroup

K of G and also G/K are simple groups of composite order and the centralizer

of

K in G is I, so

that

G may

be regarded

as a

subgroup

of

the group

of

auto-

morphisms of K. For any given prime divisor q of (K), there exist S

m

-sub-

groups

of G

which

do not

leave invariant

any

Sylow q-subgroup

of K. G

satis-

fies C

mg

but not D

mq

.

A closer analysis of the properties which a hypothetical -nr-exceptional

group

G

must have, particularly those resulting from (iii) above, would no

doubt be of some interest. Here we merely stat e:

THEOREM C4.

Let K be normal in G and suppose that both K and G/K

satisfy C

m

. Let T be an S

m

-subgroup of K, M its normalizer in G and

N = KnM. Then M/N ~ G/K. If S/N is an S

m

-subgroup of M/N, then

either G satisfies C

m

or else S/T involves a m-exceptional group.

In view of Theorem A5, this result contains both Theorems Cl and C2

as special cases. I t is a corollary of Theorem C4 th at, if

G

has a series

whose factor groups all satisfy

C

m

,

then either

G

satisfies

C

m

or else

G

involves a -or-exceptional group.

As Zassenhaus remarks (loc. cit.) the Conjectures C3 and C3* would

follow from either of the long-standing conjectures:

(a)

All groups of odd order are soluble.

(6) The group of outer automorphisms of a finite simple group is soluble.

They would also follow, as is shown by Theorem A5, from the conjecturef

(c) If A is a subgroup of the group of automorphisms of the finite group G

and if

(A)

and

(G )

are coprime, then for every prime

q

dividing

(G),

A

leaves

invariant some Sylow ^-subgroup of

G.

That (c) is true if either

A

or G is soluble follows easily from Zassenhaus's

f That the conjecture C3* is equivalent to (c) is proved by D. G. Higman, Pacific

J.

Math. 4 (1954), 545-55. I am indebted to the referee for this reference.


9/19

294 P . HALL

theorems

D6 and

D 7.

I n

this connexion

it may be

of some intere st

to

point

ou t

th e

following consequence

of

Theorem

D 5.

T H E O R E M

A6.

If 0 is a finite group satisfying E^ and A is a soluble

m-subgroup of the group of automorphisms of G, then A leaves invariant some

S

m

-subgroup of G.

For we can take G in its regular representation and regard AG as a

subgroup of the holomorph of G. By Theorem D5, A G satisfies D

m>

since

AG/G r^j A and so satisfies D^, while G satisfies

E%.

Hence A is contained

in some #

OT

-subgroup HoiAG. Therefore A leaves invariant the ^- su bg ro up

H n G of G.

We remark tha t, if A is a soluble subg roup of the group of autom orphism s

of the arbitrary finite group G and if there is at most one prime q which

divides both (A ) and (G), then we can take vr in Theorem A6 to consist of

the prime divisors of (^4) together with

q

if necessary. In this case then ,

A

leaves invariant some Sylow ^-subgroup of

G,

no special assumption

about

G

being required.

W e hav e already n oted (Corollary E2.3) th at m-serial groups satisfy

E

m

.

I t w ould follow from Conjecture C3* th a t they even satisfy D

w

. In fact

we have:

T H E O R E M D 8. Let G be a nr-serial

group.

If L is a soluble m-subgroup

of G and if H is an S

m

-subgroup of G, then L


10/19


We may therefore suppose that K is nilpotent and have only to prove

the existence of a -nr-subgroup S such that KS = G. If there is a normal

subgroup

K

x

of G such tha t 1 <

K

x

<

K,

we apply the induction h ypothesis

to G/K

x

and obtain a subgroup S

x

^ K

x

such that S

x

/K

x

is a tr-group and

KS

X

= G. Then we apply the induction hypothesis to S

x

and obtain a

-nr-subgroup S such that K

x

S = S

x

. This gives KS = KK

X

S = KS

X

= G

as required.

Finally, let K he a, minimal normal subgroup of G. Being nilpotent, it

is an Abelian #>-group. If

p

is in -nr, we can take

S

=

G.

If

p

is in -a/,

then K is a norm al Abelian /S^-subgroup of G. B y Theorem D2, G satisfies

E

m

and we can take for

S

any /S^-subgroup of

G.

Proof of

A3. We begin with an obvious rem ark. Let

G=G

0

>G

x

>...>G

r

=l

be any series of subgroups of G. Let H and K be subgroups of G such

Then

hat , for each

i

= 1,2

G = HK.

For, by hypothesis,

,...,

r,

e i ther G

e i the r G

r

_

x

^

r

t

_

x

0, that

G

t

^ J?iT. If G ^ ^ ^JEf,

then also G^- ̂ < ^ ^ , since ^ is normal in

G^^

Hence

O

t

_

x

G

x

> ... >

G

r

=

1 be a chief series of

G,

so that each

G

€

is norm al in

G.

Ev ery factor group G^JGi is a direct produ ct of isomorphic

c.f. of

G.

Hence we may divide these

r

factor groups into two mutually

exclusive classes, putting in the first class those which are TO-

0

xo-j-groups

(including those, if any, which are xn-

0

-groups) and in the second class all

the rest. We now show th at th ere is a m

0

to^-subgroup H and a TO-

0

'OT

2

-

subgroup

K

in G such tha t (i) if

G

i

_

x

lG

i

is in the first class, the n G

{

_

x

^ G

i

H,

while K^JKi is nilp ote nt, and (ii) if G

i

_

x

IG

i

is in th e second class, th en

G^

x

^

G

t

K

and

H^JHt

is nilpotent. Here we have written

H

i

= G

i

C\ H

and

K

i

=

6^ n

K.

Arguing by induction on r, we m ay assume the ex istence

of subgroups

H

and

K

containing

G

r

_

x

and such that

H/G

r

_

x

is a vr

0

m

x

-

group, K/G

r

_

x

is a

vr

0

nr

2

-group and (i)* if G^JGj is in the first class and

i

<

r,

then

G

t

_

x

<

G^

while

K

i

_

x

IK

i

is nilp ote nt, (ii)* if G^JGi is in the

second class and i


11/19

296 P. HALL

now

G

r

_

1

/G

r

~ G

r

_

x

is in the first class, we take H = H, while for K we

take a vr

Q

-nrg-subgroup of K such that

G

r

_

x

K

= K and G

r

_

x

n K is nil-

potent. Such a subgroup K exists by Theorem El*. Then it is clear from

(i)*

and (ii)* that

H

and

K

satisfy (i) and (ii) for each

i =

1,

2,..., r.

Also

H

=

H

is a xtr

0

TDygroup.

Similarly if

G

r

_

x

/G

r

is in the second class.

The remark made above shows that

G = HK.

The intersection

is a TD-

0

-group since vr

Q

, m

v

and vr

2

are mutual ly exclusive. If D

i

= G

t

C\ D,

then

D

i

_

1

ID

i

~ G

i

Di-JG^ which is a subgroup of both G

i

H

i

_

1

IG

i

~ H^JH^

and of G

i

K^JGi ~ K^JK^ But for each i, one of the two groups H^JHi and

Ki.JKj is nilpotent. Hence all the factor groups D

i

_

1

/D

i

are nilpotent

and D is soluble. Thus A3 is completely proved.

2 . 3 . LEMMA 1.

Let K be a normal subgroup of G and let H be an S

m

-

subgroup of G. Then K

n

H is an S

m

-subgroup of K and KH/K is an

S

m

-subgroup of G/K.

For let (G:K) = h and (K) = k. Then

{G)

m

= h

m

k

m

= (H) = (H:Kn H)(K

n

H).

But

H/K

n

H ~ KH/K,

which is a w-subgroup of

G/K,

so that

(H:Kn H)

divides h

m

\ and K n H is a To--subgroup of K so that (K n H) divides k

m

.

Hence (KH:K) = h

m

and (K n H) — k

m

, as required.

Proof of E2. Let K satisfy C

m

and let 6?/Z satisfy E

m

. Let T be an

/S^-subgroup of

K

and let

N

be its normalizer in

G.

For any

x e G, T

x

is

an /S^-subgroup of iT and hence is conjugate to T in K so that T

x

= T

v

with y e . Thus

xy-

1

e N

and

NK = G.

Hence JV/Z n

N

~

G/JK" SO

that

JV/X n JV has an #

OT

-subgroup H/K n JV. Apply Theorem El to the group

H/T with the normal ̂ --subgroup K n

iV/2

7

and we obtain an ^-subgroup

U/T of H/T. Then (U:T) = {H.K D N) = h

m

and (T) = /c

OT5

so that U

is an /S^-subgroup of G.

L E M M A

2. .Le£

H = H

x

xH

2

x

...xH

r

.

If each H

t

has a given one of the

properties E

w

, C

w

, D

m

, E*

m

, C

s

m

, D

s

m

, E , or C^, then H has the same property.

The proof is straightforward and may be omit ted .

Suppose now that all the c.f. of G satisfy C

m

, and let K be a minimal

normal subgroup of G. Then K is a direct product of c.f. of G. Hence K

satisfies C

m

by Lemma 2. Corollary E2.1 now follows from Theorem E2

by an induction argument on (

0),

since we may assume tha t

G/K

satisfies

E

m

.

Since trr-groups satisfy C

m

trivially and

TO-'^-groups

satisfy C

m

by Sylow's

theorem, we obtain Corollary E2.2 as a particular case of E2.1.

As for Corollary E2.3, this follows from E2.2 for

m

1

= w, m', vrq,

and


12/19


13/19

298 P . HALL

We make a few pre lim inar y simplifications. Since G/K satisfies D

m

, we

have L


14/19


Now let

Q

be the normalizer of

P

in

G

and let

Q

=

K

n

Q.

We first

prove (5) that Q satisfies El^. Since P and Q are both normal in Q and

Pn Q ^. Lr\ K = I,

we have

PQ = PxQ.

Let the distinct primes

dividing (G)^ be p = p

x

, p

2

,..., p

r

and let Q

i

be a Sylow

r

subgroup of Q

for

i —

1, 2,..., r. Since T satisfies Z^ we may choose an iS^-subgroup T$

of T such tha t P x (^ < T,. We note that ^ = Z n 2J is an ^ -s ub gro up

of̂ fiT by Lemma land

is

therefore nilpotent. Also

T

= i£ P and

so 7^

=

S

i

P.

Hence T

t

contains a unique Sylow ^-subgroup P^ containing P and the

centralizer

Z

t

of

P

t

in

T

t

is nilpotent and contains

Q^

Since

T

satisfies

D

mi

the subgroups T̂ are all isomorphic. More precisely, there is an isomorphism

mapping

T

t

on to

T

x

and at the same time mapping

P

t

on to

P

x

and therefore

Zi onto Z

x

. If this isomorphism maps Qi onto ii^for i = 2,

3,...,

r, itfollows

that iẐ <

Z

x

.

Since

Z

x

is nilpotent, the product

R

of

R

x

= Q

x

,

R

2

,...,

R

r

is direct; and since P -group,

P

< P

x

would imply that

P < $ n

P

lt

so that

P = LOT

would not be an ^- su bg ro up of

Q

n T

7

and therefore, by Lemma 1, L would not be an iS^-subgroup of Q. Hence

P =

P

x

is a Sylow ^-subgroup of

H

n T. Also

H 0 T is

normal in J?. By

Sylow's theorem, the normalizer Q C\ H = U of P in H satisfies H ^. TU.

Since T =

KP

and P <

U,

we even have

H ^KU.

Since

KH = G by

(2), we have i£?7 = £. But LnK = 1. Hence (17) > (X). But £ is an

/S^-subgroup of

Q

and

?7

is a -or-subgroup of

Q.

Hence

U

is an /S^-subgroup

of Q. But Q satisfies D

m

by induction. Hence L = U

x

^ ^

x

for some

x e G

and Theorem D5 is completely proved.

Corollary D5.1 follows immediately. If G has a series all of whose factor

groups satisfy

E^,

then all the c.f. of

G

satisfy E

1

^ by Lemma 1. If

K

is

a minimal normal subgroup of G, then K satisfies E

1

^. by Lemma 2. The

c.f. of

G/K

all satisfy

E%

and we may suppose inductively that

GjK

satisfies

D

s

m

.

Then

G

satisfies

D%.

by Theorem D5.

Corollary D5.2 is a particular case of D5.1. For if

G

is -nr-separable and

vr

1

is any non-empty subset of

w,

then every c.f. of

G

is a

w'

x

p

-group for

some prime

p

in

-m

x

and therefore satisfies

E^

by Sylow's theorem.

Corollary D5.3 is a special case of D5.2.

Proof of A1. Suppose H and K are S

m

-subgroups of G and that both H

and

K

have Sylow series of complexion

(p

1

,P2>-'->P

r

)>

where

p

x

,..., p

r

are the

distinct primes dividing (G)

m

. If H

x

and K

x

are the normal

S

VlPi

Prl

-

subgroups of

H

and

K,

respectively, then

H

x

and

K

x

have Sylow series of


15/19

300 P. HA LL

complexion (Pi,.--,Pr-i)

a n

d

a r e

$p

lf

2>,

3

»

r

_

1

-subgroups of

G.

Arguing by

induction on r, we may suppose

H

x

and

K

x

are conjugate in

G.

We may

then take

H

x

= K

x

,

replacing

K

by a conjugate if necessary.

H/H

x

smd.K/H

x

are then two Sylow ̂ -su bg rou ps of

N /H

x

,

where

N

is the normalizer of

H

x

in

G.

Hence

H

is conjugate to

K

in

N.

2.5. Proof of

D7*. I t will be sufficient t o show th a t a ttr'-soluble grou p

G

satisfies

D

m

.

Let

K

be a minimal normal subgroup of G. Then

K

is either a -nr-group

or else an Abelian g-group for some prime

q

in

m'.

Also

G/K

is icr'-soluble.

Arguing by induction on

(G),

we may assume th at

G/K

satisfies

D

m

.

If

K

is a -nx-group, then

G

satisfies

D

m

by Lemm a 4. Thus we may assume t ha t

K

is an Abelian -nr'-group.

G

satisfies

E

w

by Theorem E2. Let

H

be an

^ - s u b g r o u p o f

G

and

L

any xtr-subgroup of

G.

Since

G/K

satisfies

D

m

by

the induction hypothesis, we have L


16/19


K cannot be Abelian or G would satisfy C

m

by Theorem D2. Hence

K

=

K

1

xK

2

X...xK

r

,

where the

K

€

are isomorphic simple groups of com-

posite order. The

r

subgroups iQ are the only normal subgroups of

K

which are simple. On transforming K by the elements of H, the K

t

must

therefore be permuted transitively among themselves. Hence if N is the

normalizer of

K

x

in

G,

we have

K

<

N

and

(G:N) = r.

Let

q

be a prime

divisor of (Kj) and let Q

x

be any Sylow (/-subgroup of K

x

. Let N be the

normalizer of Q

x

in N. By Sylow's theorem, K

X

N = N and therefore

KN = N. Hence N/K n N ~ F / Z and, by Theorem El , 2V contains an

£

OT

-subgroup

U,

so that

{K n N)U = N

and therefore

KU = N.

Thus £7

is an iS

m

-subgroup of N. Since 6r = iT£T = KH

X

and K ^. N, the inter-

sections V = H r\ N and 1̂ = T̂

x

n iV are also ^. -subgroups of N.

Suppose if possible that

r

> 1. Then

N < G

and hence

N

satisfies

C

m

.

Consequently

U, V, V

x

are conjugate in

N.

Replacing / / and

H

x

by con-

jugates, we may assume tha t U = V = V

x

. We then have (H:U) = r and

may choose the elements h

x

= 1, h

2

,..., h

r

in H so that

. £*

= jfî (i = i, 2,..., r).

Then Qi* = Qi is a Sylow ^-subgroup of

iQ

and hence

Q = Q

1

xQ

2

X...xQ

r

is a Sylow ^-subgroup of K. For any xe H, we have ^x = u^-, where

w

e

C7

and j = j (i ,3 ). Hence Qf =

Q^

x

= Q * = Q$ = Q

p

since

U

<

N.

For a fixed xe H, i -+j(i,x) is a permutation of 1, 2,..., r. Hence Q

x

= Q

and so i/ is contained in the normalizer R of Q in 6r. Similarly H

x

is con-

tained in R. Since $ is not normal in K, we have R < G and so 72 satisfies

C

m

. Hence the ^-s ubgro ups H and H

x

of i? are conjugate, a contradiction.

Thus r = 1 and i£ is a simple -nr'-group of composite order.

By Sylow's theorem

KR = G,

so tha t the ^-subgroups of

R,

which are

all conjugate in R as we have already remarked, are also ^- su bgr ou ps

of

G.

There is therefore a uniquely determined class of conjugate

^-sub group s of G associated with any given prime divisor q of (K), viz. those

which belong to the normalizer in

G

of some Sylow (/-subgroup of

K

. Since

G does not satisfy C

m

, there must be some S

m

-subgroups of G which do

not leave invariant any Sylow (/-subgroup of

K.

If

H

is one of these, then

H cannot be contained in any

m(2

-subgroup L of G, for L n K is normal

in

L

and is a Sylow (/-subgroup of

K.

Thus

G

cannot satisfy

D

mq

.

But

G satisfies E

mq

bj Corollary D5.3, or more simply by Sylow's theorem and

Theorem El. In fact, G satisfies C

OT(r

For let L and L

x

be any two

/S^-subgroups of G. Then LP\K and L

X

D K are Sylow (/-subgroups of K and

therefore conjugate. Replacing L

x

by a conjugate if necessary, we may

suppose that Lc\ K = L

x

n K = Q, so that both L and L

x

belong to the

normalizer R of Q. L/Q and LJQ are then ^-s ubgrou ps of R/Q and


17/19

302 P. H A L L

therefore conjugate because R/Q is properly involved in G and so satisfies

C

m

.

Hence

L

and

L

x

are conjugate.

Let Z be the centralizer of K in

G.

Then Z is norm al in 0 and K n Z = 1

because if is simple bu t not cyclic. Since G/K is also simple, Z ^

1

would

imply that G = KxZ. This would make Z a normal and therefore the

unique ^-subgroup of G. This is impossible since G does not satisfy C

w

.

Hence

Z

= 1 and Theorem A5 is completely proved .

Proof of C4. Let K be a normal subgroup of G such that both K and

#/ £ satisfy C^. By Theorem E2,

G

satisfies

E

m

.

Suppose th at G does not

satisfy

C

m

and let

H

and

H

x

be

/S .̂

-subgroups of

G

which are not con-

jugate in G. The ^ - subgroups H n K and H

X

C\ K of K are conjugate and

we may assume H f) K = H

X

C\ K = T, replacing H

x

by a conjugate if

necessary. Then both

H

and

H

x

are contained in the normalizer

M

of

T

in G. Let N = K n M. Since Z satisfies C

OT

, we have KM = G as in the

proof of Theorem E2 . Hence M/N ~ £ / Z and satisfies C

w

: The

^ - subg roups NH/N and NHJN of Jf/iV are therefore conjuga te and we

may assume that

NH = NH

X

=

S,

replacing

H

x

by a conjugate if neces-

sary. S/T has the normal ̂ - su b gr ou p N/T and has the two non -conjugate

S^-subgroups

H/T

and

HJT.

Hence

S/T

does no t satisfy

C

m

.

Therefore

S/T

involves a w-exceptional group and Theorem C4 is prove d.

2.7. Proof of D8. We argue by induction on (G). Let K be a minimal

normal subgroup of

G, H

any /8^-subgroup of

G,

and

L

any xo--subgroup

of G. We have to prove that, if either L is soluble or if G involves no

•nr-exceptional group, then

L

^

H

x

for some

x e G.

Since

KL/K

is a

-nr-subgroup and KH/K is an /S^-subgroup of G/K, we may suppose by

induction that

L

<

KH

X

for some

x

e

G;

for if

L

is soluble, so is

KL/K,

while if (3 involves no xir-exceptional group, neither does G/K. Also, KH

X

is a -nr-serial group and involves no -nr-exceptional group unless G involves

one.

The result now follows by induction if KH

X


18/19

T H E O R E M S L I K E S Y L O W ' S

303

2.8. Proof

of A4. Let p < q

<

w, where p

and

q

are

primes,

and let

H

be an $

pg

-subgroup of the sym metric group S

n

. By a theorem of Burn side,

H is

soluble.

Suppose first that H is a primitive permutation group. Then n = r

m

,

where r is

a

prime, an d r is either p

or q.

Also, H is contained

in

the holo-

morph H of the elementary Abelian group of order r

m

. Cf. Speiser (13,

Satz 98). Since

(H)

r

= r^-

m(

-

m+x

>

and

(H)

r

= ( SJ

r

=

r

i+r+r*+...+r»-»

} w e m u s t

h a v e

£

This equation implies that either m =

1

or else

m =

2,

r =

2. In the case

m

=

1,

if is the

metacyclic group

of

order r(r—1),

so

tha t

we

must have

r =

q = n =

l-\-kp. But then (?>

n

)

p

^

p

k

,

so t ha t

&

must be divisible by

p*-

1

.

Thus either k =

2,

p =

2, or

k.=

1. But

(S

5

)

2

= 8 so

tha t &

=

2

is

impossible. Hence jfc=l,^>

= 2, g = 3, and

H =

S

3

.

If, on the other hand, m = 2, so th at r = 2 = p, we have w = 4 and so

q = 3, H = S

4

. We conclude tha t the only cases where H can be primitive

are when p = 2, q = 3, and w = 3 or 4, and in these cases H is the whole

of S,

/r

In all other cases

H

is either imprimitive or intransitive.

Ne xt suppose th at H is intran sitive , with com ponents of degrees

I

and m

where £+ra

=

n.

Then

i/ is

contained

in S

z

x

S.

m

and, since

it

must be

an

$p

S

-subgroup

of

this group, both

£j and S

m

must satisfy

JE^.

Further,

# cannot exceed both I and m, since the n (H ) would be prime to q, contrary

I

of

S; X S

wl

in S

M

must

be

prime

to

pq,

mj

since otherwise H could

not be an

$p

g

-subgroup

of

S

M

.

A similar argument applies

if

JÊ

is

imprimitive, with

e

systems

of

imprimitivity of d symbols each. Then n = de and H is contained in

S^ISg,

a

maximal imprimitive subgroup

of S

M

corresponding

to

this

factorization of n. The order of 2^1; £

e

is (d\)

e

e\ and therefore q cannot

exceed both d

and e.

Since G

=

S

d

1 S

e

has a

normal subgroup K which

is the direct prod uct of e factors isomorphic with 2

d

and G/K ~ S

e>

it is

clear from Lem ma 1 th at bo th

2

d

and 2

e

must satisfy E

pq

. Further

the

index n\/(d\)

e

e\ of G in S

n

must be prime to pq.

If therefore JET

is

either imprimitive

or

intransitive, there

is

necessarily

an integer n

x

such that q


19/19

304 T H E O R E M S L I K E S Y L O W ' S

A subgroup

of

index

35 in S

7

c a n n o t

be

t r ans i t ive

and so

m u s t h a v e

the form E

3

x S

4

. Again

we

h a v e

a

single class

of

#

2 3

- subgroups . None

of

these con ta ins

a

cyc l ic pe rmuta t ion

of

order

6.

T h u s

S

7

satisfies

C

23

but

n o t D

2

3

.

A subgro up of index 35

in S

8

c a n n o t

be

in t ran s i t ive s ince

all th e

b inomia l

coefficients 8, 28, 56 ,70 ar e even .

I t

cann ot be pr im it ive s ince th e ho lom orph

of

the

e le m e n ta r y g r o u p

of

order

8 has

index

30 in S

8

.

H e n c e

it

m u s t

be

im p r im i t i v e

and

on ly

the

subgroups

of the

form S

4

1 S

2

are

la rge e noug h.

Aga in

we

h a v e

a

single class

of

conjugate

S

2

3

-subgi\oups. None

of

t h e m

con ta ins

a

s u b g r o u p

of

th e form 2

2

1 E

4

. Thus

2

8

satisfies C

2 3

bu t not

Z )

23

.

I t r ema ins

to

show tha t ,

if n > 8, E

n

does

not

satisfy

E

2 3

. Suppose

the

c o n t r a r y

and

choose

n > 8 as

smal l

as

possible

so

t h a t

2

n

has an

S

2 3

- subgroup H. If H is in t rans i t ive w i th componen ts of degrees I and m,

w h e r e I

d

%

S

e

where

de

= n. As we have

also seen, both 2,

d

and 2

e

must have $

23

-subgroups and n\/(d\)

e

e\ must be

prime to 6. By our choice of n, both d and e must be ^ 8, ^ 6, and ̂ 1.

But in this case H is transitive so that n must be of the form 2

a

3^. Thus

d

and e can only be 2, 3, 4, or 8. Again it is easy to verify that none of the

eleven relevant indices

is

prime to

6.

This concludes the proof of Theorem

A4.

REFERENCES

1.

W. BURNSIDE, Theory

of

groups

of

finite order, 2nd edition (Cambridge, 1911).

2.

S.

A.

CUNIHIN,

Mat. Sbornik,

N .S . (25) 67

(1949), 321-46.

3.

Doklady

Akad.

Nauk S.S.S.R.

N.S.

73

(1950), 29-32.

4. ibid.

95

(1954), 725-7.

5. ibid.

66

(1949), 165-8.

6. ibid.

69

(1949), 735-7.

7. Mat. Sbornik, N.S.

(33) 75

(1953), 111-32.

8.

P.

A.

GOL'BERG, Doklady

Akad. Nauk S.S.S.R.

N .S . 64

(1949), 615-8 .

9.

P.

HAXL,

J. London Math. Soc. 3

(1928), 98-105.

10. Proc. London Math. Soc. (2) 43

(1937), 316 -23.

11. J . London Math. Soc. 12

(1937), 198-200.

12. and GRAHAM HIGMAN, Proc. London Math. Soc. (3) 6

(1956),

1-42.

13.

A.

SPEISER,

Theorie der Gruppen

von

endlicher Ordnung,

3rd

edition (Berlin,

1937).

14.

H. WIELANDT,

Math. Zeitschrift,

60

(1954), 40 7-8 .

15.

H. ZASSENHAUS,

Lehrbuch der Gruppentheorie (Leipzig and Berlin, 1937).

King's College,

Cambridge

theorems like sylow's (hall)

Documents