on automorphisms fixing certain groups

December 7, 2012 11:58 WSPC/S0219-4988 171-JAA 1250163

Journal of Algebra and Its ApplicationsVol. 12, No. 2 (2013) 1250163 (17 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0219498812501630

ON AUTOMORPHISMS FIXING CERTAIN GROUPS

ZAHEDEH AZHDARI∗ and MEHRI AKHAVAN-MALAYERI†,‡

Department of Mathematics, Alzahra UniversityVanak, Tehran, 19834, Iran∗z−[email protected]†[email protected]‡[email protected]

Received 21 December 2011Accepted 1 May 2012

Published 11 December 2012

Communicated by S. Sehgal

Let G be a group and let M and N be two normal subgroups of G. Let AutMN (G)

be the set of all automorphisms of G which centralize G/M and N . In this paper,we find certain necessary and sufficient conditions on G such that AutM

N (G) be equalto Z(Inn(G)), Inn(G), C∗ or Autc(G). We also characterize the subgroups of a finite

p-group G for which the equality AutM1N1

(G) = AutM2N2

(G) holds.

Keywords: Inner automorphism; central automorphism; nilpotent group.

Mathematics Subject Classification: Primary: 20D45; Secondary: 20E36

0. Introduction

Let G be a group. An automorphism α of G is called central if x−1α(x) ∈ Z(G)for all x ∈ G. The set of all central automorphisms of G, denoted by Autc(G), is anormal subgroup of Aut(G). We denote by CAutc(G)(N) and C∗ = CAutc(G)(Z(G)),the group of all central automorphisms of G fixing N and Z(G) elementwise, respec-tively.

Let M and N be two normal subgroups of G. By AutM (G), we mean thesubgroup of Aut(G) consisting of all the automorphisms which centralize G/M andby AutN (G), we mean the subgroup of Aut(G) consisting of all the automorphismswhich centralize N . We denote AutM (G) ∩ AutN (G) by AutM

N (G). We recall thatC∗ = AutZ(G)

Z(G)(G) and Autc(G) = AutZ(G)G′ (G).

It is interesting and natural to discuss the question of “finding the necessaryand sufficient conditions for a group G such that certain subgroups of Aut(G) beequal”.

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http://dx.doi.org/10.1142/S0219498812501630


Z. Azhdari & M. Akhavan-Malayeri

There are some well-known results in this regard for a finite group G. Curranand McCaughan in [6] and Curran in [5] characterized finite non-abelian p-groupsG for which Autc(G) = Inn(G) and Autc(G) = Z(Inn(G)), respectively.

In [2], Shabani Attar characterized all finite p-groups G for which the equalityC∗ = Inn(G) holds.

Yadav in [9] obtained some necessary and sufficient conditions for the equalityAutc(G) = C∗ on p-groups of class 2 and Jafari in [7] omitted the class restrictionon the groups and he characterized all finite p-groups with this property.

Our first motivation for this paper is to study the problem in infinite groups.For infinite groups the situation is more complicated. In [4], we tried to characterizefinitely generated groups for which C∗ = Inn(G). In this paper we investigate whenAutM

N (G) is equal to Z(Inn(G)), Inn(G), C∗ or Autc(G) in special cases. Thenmore generally we find solutions for the equation AutM1

N1(G) = AutM2

N2(G) in a finite

p-group G.In the following theorem, first we characterize the subgroups of a group G for

which the inclusion Z(Inn(G)) ≤ AutMN (G) holds. This inclusion possesses obvious

solution for M = Z(G) and N = G′. Then we discuss the equalities Z(Inn(G)) =AutM

N (G) and Inn(G) = AutMN (G) in special cases.

Theorem A. Suppose N is a normal subgroup of a non-abelian group G. And letM be a central subgroup of G where Z(G/M) = K/M . Then

(i) Z(Inn(G)) ≤ AutMN (G) if and only if K = Z2(G) ≤ CG(N).

(ii) If M ≤ N and Z2(G)/Z(G) is finite then Z(Inn(G)) = AutMN (G) if and only

if K = Z2(G) ≤ CG(N) and Hom(G/N, M) � Z2(G)/Z(G). In particularInn(G) = AutM

N (G) if and only if N ≤ Z(G), G′ ≤ M and Hom(G/N, M) �G/Z(G).

We will show that the finiteness Z2(G)/Z(G) in part (ii) is necessary.

As immediate consequences of Theorem A, we obtain the main theorem of Sha-bani Attar in [2] and one of the main theorems of Curran in [5] and Curran andMcCaughan in [6].

Let G be a finite nilpotent group. Then G is the direct product of its Sylowp-subgroups, say G � Gp1 × · · · × Gpn . So AutM

N (G) � AutMp1Np1

(Gp1) × · · · ×AutMpn

Npn(Gpn) where Mpi and Npi are Sylow pi-subgroups of M and N , respec-

tively. Therefore, without loss of generality, from now on we restrict attention tothe case where G is a p-group.

Let G be a finite p-group and M1 and M2 be two central subgroups of G. Then

M1 = Cpa1 × Cpa2 × · · · × Cpas ,

M2 = Cpb1 × Cpb2 × · · · × Cpbs′ ,

where a1 ≥ a2 ≥ · · · ≥ as > 0 and b1 ≥ b2 ≥ · · · ≥ bs′ > 0 are cyclic decompositionsof M1 and M2.

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On Automorphisms Fixing Certain Groups

Let N1 and N2 be normal subgroups of G. Using the above notations, thesubgroups of G for which the equality AutM1

N1(G) = AutM2

N2(G) holds, will be char-

acterized.

Theorem B. Let G be a finite p-group. Let M1, M2, N1 and N2 be normal sub-groups of G such that Mi ≤ Z(G) ∩ Ni for i = 1, 2, M1 ≤ M2 and N2 ≤ N1. ThenAutM1

N1(G) = AutM2

N2(G) if and only if one of the following statements holds :

(i) M1 = M2 and N1 ≤ G′Gpn

N2 where exp(M1) = pn, or(ii) N1 = N2, s = s′ and exp(G/G′N1) ≤ pat where t is the smallest integer between

1 and s such that aj = bj for all t + 1 ≤ j ≤ s.

Notice that the conditions “M1 ≤ M2 and N2 ≤ N1” may be omitted (see Corollary2.17). Also the conditions “Mi ≤ Z(G) ∩ Ni for i = 1, 2” may be substituted by “G

be a purely non-abelian group”.

As a consequence of Theorem B, we obtain necessary and sufficient conditionsfor the subgroups of a finite p-group G for which the equality AutM

N (G) = C∗ orAutM

N (G) = Autc(G) holds. We will use the following notation.Let G be a finite p-group. Let M be a central subgroup of G. Then we may

write

M = Cpa1 × Cpa2 × · · · × Cpas ,

Z(G) = Cpb1 × Cpb2 × · · · × Cpbs′ ,

where a1 ≥ a2 ≥ · · · ≥ as > 0 and b1 ≥ b2 ≥ · · · ≥ bs′ > 0. Let t be the smallestinteger between 1 and s such that aj = bj for all t + 1 ≤ j ≤ s (else t = s). Also ifG is a nilpotent group of class 2 then we may write

G/Z(G) = Cpc1 × Cpc2 × · · · × Cpcr ,

G/G′ = Cpd1 × Cpd2 × · · · × Cpdr′ ,

where c1 ≥ c2 ≥ · · · ≥ cr > 0 and d1 ≥ d2 ≥ · · · ≥ dr′ > 0. Let k be thelargest integer between 1 and r such that c1 = · · · = ck = c. Set H = H/Z(G) =Cpc1 × Cpc2 × · · · × Cpck and K = K/G′ = Cpd1 × Cpd2 × · · · × Cpdk .

By using the above notation we have the following corollary.

Corollary C. Let G be a finite non-abelian p-group. Let M and N be two normalsubgroups of G such that M ≤ Z(G) ≤ N . Then

(i) AutMN (G) = C∗ if and only if one of the following statements holds :

(a) M = Z(G) and N ≤ G′Gpn

Z(G) where exp(Z(G)) = pn, or(b) N = Z(G), s = s′ and exp(G/G′Z(G)) ≤ pat.

(ii) AutMN (G) = Autc(G) if and only if one of the following statements holds :

(c) M = Z(G) and N ≤ G′Gpn

where exp(Z(G)) = pn, or(d) N ≤ G′, s = s′ and exp(G/G′) ≤ pat.

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In particular if G is nilpotent of class 2 then(iii) AutM

N (G) = C∗ if and only if M = N = Z(G) or N = Z(G), s = s′ andexp(G/Z(G)) ≤ pat .

(iv) AutMN (G) = Autc(G) if and only if M = N = Z(G) = G′Gpn

whereexp(Z(G)) = pn if and only if M = N = Z(G), r = r′, (G/Z(G))/H �(G/G′)/K and exp(Z(G)) = exp(G′).

1. Preliminary Results

Our notation is standard. Let G be a group. By Cm, G′, Z(G), T (G), Aut(G) andInn(G), we denote the cyclic group of order m, the commutator subgroup, thecenter, the torsion group, the group of all automorphisms and the group of allinner automorphisms of G, respectively. Let Gpn

= 〈gpn

: g ∈ G〉. For any group H

and abelian group K, Hom(H, K) denotes the group of all homomorphisms fromH to K. We say that a non-abelian group G is purely non-abelian if G has nonon-trivial abelian direct factor.

Adney and Yen in [1] introduced the one-to-one map θ : Autc(G) →Hom(G/G′, Z(G)) defined by θ(α) = α∗ where α∗(gG′) = g−1α(g) for eachα ∈ Autc(G) and g ∈ G. Also they proved that if G is a purely non-abelian finitegroup, then there exists a bijection between Autc(G) and Hom(G/G′, Z(G)).

Let G be a group and M and N be two normal subgroups of G. Here we establisha bijection between AutM

N (G) and Hom(G/N, M).

Lemma 1.1. Let G be a group and M and N be two normal subgroups of G suchthat M ≤ Z(G). Let θ : AutM

N (G) → Hom(G/N, M) defined by θ(α) = α∗ whereα∗(gN) = g−1α(g) for each α ∈ AutM

N (G). Then

(i) θ is one-to-one.(ii) If G is purely non-abelian and satisfies max-n and min-n then θ is onto.

In particular if G is a finite purely non-abelian group then |AutMN (G)| =

|Hom(G/N, M)|.

Proof. Since α is an automorphism fixing N elementwise, α∗ is a well-definedhomomorphism of G/N to M . Therefore θ is a well-defined map.

(i) Clearly, θ is one-to-one.(ii) Let β ∈ Hom(G/N, M), then α : G → G defined by α(g) = gβ(gN) is a

homomorphism centralizing G/M and N . By using an argument similar toMuller’s paper [8], we show that α is an automorphism and this concludes theproof of (ii). Since M is a central subgroup, α is a central endomorphism, henceG = kerαk × Imαk for some positive integer k. Now induction on r shows thatkerαr ≤ Z(G). Therefore kerαk is an abelian direct factor of G. But G ispurely non-abelian, hence kerαk = 1. Therefore kerα = 1 and Imα = G orequivalently α is a bijection, as required.

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In [4] we proved that if Z(G) ≤ G′ then Autc(G) � Hom(G/G′, Z(G)). Recallthat Autc(G) fixes G′ elementwise. In fact, by substituting G′ by N and Z(G) byM , we have a similar result for AutM

N (G). Indeed, in any group G if M ≤ Z(G)∩N

then AutMN (G) � Hom(G/N, M). Therefore the following lemma generalizes the

Proposition 1.5(i) of [4].

Lemma 1.2. Let G be a group and M and N be two normal subgroups of G suchthat M ≤ Z(G) ∩ N . Then AutM

N (G) � Hom(G/N, M). In particular AutMN (G) is

abelian.

Proof. Let θ be the correspondence α → α∗ defined above. In the first place thisis a homomorphism: for if α1, α2 ∈ AutM

N (G) and g ∈ G, then

(α1α2)∗(gN) = g−1α1α2(g) = g−1α1(α2(g))

= g−1α1(gg−1α2(g)) = g−1α1(g).α1(g−1α2(g))

= g−1α1(g).g−1α2(g) = α∗1(gN)α∗

2(gN).

Clearly, by Lemma 1.1, θ is one-to-one. Our homomorphism is also surjective, forthis let β ∈ Hom(G/N, M), we define the map

α : G → G,

g → gβ(gN)

evidently α is a well-defined homomorphism. By [4, Lemma 1.1], α is an isomor-phism. Furthermore α centralizes G/M and N and consequently α ∈ AutM

N (G).Also by the definition of θ, α∗ = β and it follows that θ is an isomorphism ofAutM

N (G) to Hom(G/N, M), as required.

The following lemma is crucial for our study of AutMN (G).

Lemma 1.3. Let G be a group and M and N be two normal subgroups of G. ThenAutM

N (G) ∩ Inn(G) � (K ∩ CG(N))/Z(G) where Z(G/M) = K/M . In particular,if AutM

N (G) ≤ Inn(G) then AutMN (G) � (K ∩ CG(N))/Z(G).

Proof. It is straightforward to see that g ∈ K∩CG(N) if and only if Ig ∈ AutMN (G)

where Ig is the inner automorphism induced by g. And a quick calculation showsthat the map ϕ : K ∩ CG(N) → AutM

N (G) ∩ Inn(G) defined by ϕ(x) = Ix−1 , for allx ∈ K ∩ CG(N) is an epimorphism with the kernel equal to Z(G), as required.

Other results which will be used in our proofs is stated for convenience in thefollowing lemmas.

Lemma 1.4 ([9, Lemma 2.4]). Let G be a finite p-group such that Autc(G) = C∗.Then G is purely non-abelian.

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From Lemma 1.4, we deduce that if G is a finite p-group, in which Autc(G) = C∗

then Autc(G) � Hom(G/G′, Z(G)) � Hom(G/G′Z(G), Z(G)).

Lemma 1.5 ([9, Lemma 2.5]). Let A and B be two finite abelian p-groups suchthat A = Cpa1 × Cpa2 × · · · × Cpas where a1 ≥ a2 ≥ · · · ≥ as > 0, and B =Cpb1 × Cpb2 × · · · × Cpbs where b1 ≥ b2 ≥ · · · ≥ bs > 0. Let bj ≥ aj for all j,

1 ≤ j ≤ s, and bj > aj for some such j. Let t be the smallest integer between 1 ands such that aj = bj for all j such that t + 1 ≤ j ≤ s. Then, for any finite abelianp-group C, |Hom(A, C)| < |Hom(B, C)| if and only if the exponent of C is at leastpat+1.

2. Proof of Main Results

In this section we investigate the relationship between AutMN (G) and Z(Inn(G)) in

special cases. First we prove Theorem A.

Proof of Theorem A. (i) First assume that K = Z2(G) ≤ CG(N) and g ∈ Z2(G).Observe that g ∈ Z2(G) is equivalent to Ig ∈ Z(Inn(G)). Since x−1Ig(x) = [x, g] ∈[G, Z2(G)] = [G, K] ≤ M for all x ∈ G and also by assumption [x, g] ∈ [N, Z2(G)] =1 for all x ∈ N , then Ig ∈ AutM

N (G).For the converse suppose that Z(Inn(G)) ≤ AutM

N (G). Since K/M = Z(G/M),it follows that [G, K] ≤ M ≤ Z(G). Hence [K, G, G] = 1 or equivalently K ≤ Z2(G).On the other hand, Ig ∈ AutM

N (G) for every g ∈ Z2(G) and hence [G, Z2(G)] ≤ M ,consequently Z2(G) ≤ K, that is K = Z2(G). Also [x, g] = 1 for all x ∈ N andg ∈ Z2(G), hence [N, Z2(G)] = 1 and Z2(G) ≤ CG(N). This completes the proof (i).

(ii) Suppose that Z(Inn(G)) = AutMN (G). Then by part (i), K = Z2(G) ≤

CG(N) and by Lemma 1.2, Hom(G/N, M) � Z2(G)/Z(G).Conversely, since Hom(G/N, M) � Z2(G)/Z(G), AutM

N (G) � Z(Inn(G)). Onthe other hand K = Z2(G) ≤ CG(N) implies that Z(Inn(G)) ≤ AutM

N (G). ButZ2(G)/Z(G) is finite and hence Z(Inn(G)) = AutM

N (G), as required.Now if Inn(G) = AutM

N (G) then Inn(G) ≤ C∗ and so G is nilpotent groupof class 2. Therefore K = Z2(G) = CG(N) = G and consequently N ≤ Z(G),G′ ≤ M and Hom(G/N, M) � G/Z(G). Conversely, if N ≤ Z(G) and G′ ≤ M

then Inn(G) ≤ AutMN (G) and Hom(G/N, M) � G/Z(G). So the equality holds.

It should be noted that finiteness of Z2(G)/Z(G) in the Theorem A(ii) is nec-essary. In fact if M = N = Z(G) then there exists a nilpotent group G of class 2with infinite G/Z(G) = Z2(G)/Z(G) such that Inn(G) < C∗ = AutM

N (G) and alsoHom(G/Z(G), Z(G)) � G/Z(G) (see [4, Example 3.5]).

Now we give some important consequences of Theorem A.

Corollary 2.1. Let G be a finitely generated non-abelian group and M and N

be normal subgroups of G such that M ≤ Z(G) ≤ N and G/Z(G) is finite.Then AutM

N (G) = Inn(G) if and only if G is a nilpotent group of class 2,

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N = Z(G), G′ ≤ M and M � Cm × H × Zr in which Cm � Πp∈π(G/Z(G))Mp,

H � Πp�∈π(G/Z(G))Mp and r ≥ 0 is the torsion-free rank of M .

The proof of Corollary 2.1 requires the Corollary 1.4 in [4]. First we discuss thefollowing correction.

Let G be a finite abelian group. We denote by Gp, the p-primary componentof G. Hence G = Πp∈π(G)Gp where π(G) denotes the set of all prime number p

dividing |G|. In [4] we claim the following corollary.

Corollary ([4, Corollary 1.4]). Let A and B be two finite abelian groups andexp(B) is divisible by exp(A). Then Hom(A, B) � A if and only if B is a cyclicgroup.

This result is incorrect and in Corollary 2.2 we prove correct version of thecorollary.

In fact the conditions “B is a cyclic group” should be replaced by “B � Cm×H

in which Cm � Πp∈π(A)Bp and H � Πp�∈π(A)Bp”. Now we prove this result:

Corollary 2.2. Let A and B be two finite abelian groups and exp(A)| exp(B).Then Hom(A, B) � A if and only if B � Cm × H in which Cm � Πp∈π(A)Bp andH � Πp�∈π(A)Bp. In particular, if π(A) = π(B) then this is equivalent to B is acyclic group.

Proof. Let π(A) = {p1, . . . , pk} and π(B) = {p1, . . . , pk, . . . , ps} where s ≥ k. Wemay write

A =k∏

i=1

Api ,

and

B =k∏

i=1

Bpi ×s∏

i=k+1

Bpi .

From [4, Lemma 1.2] it follows that

Hom(A, B) = Hom

(k∏

i=1

Api ,

k∏i=1

Bpi ×s∏

i=k+1

Bpi

)

�k∏

i=1

Hom(Api , Bpi).

Since exp(B) is divisible by exp(A), exp(Bi) = pbi

i is divisible by exp(Ai) = pai

i forall 1 ≤ i ≤ k. Put H = Πs

i=k+1Bpi (if s = k then H = 1). Clearly, Hom(A, B) � A

if and only if Hom(Api , Bpi) � Api for all 1 ≤ i ≤ k. Now by [4, Proposition 1.3],this is equivalent to Bpi is cyclic for all 1 ≤ i ≤ k or equivalently B � Cm × H

where m = pb11 · · · pbk

k .

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Since in [4] we used Corollary 1.4 in several places, we require the followingreplacements in [4].

Theorem 2.3 ([4, Theorem 0.1]). Let G be a finitely generated nilpotent groupof class 2. Then C∗ � Inn(G) if and only if Z(G) is infinite cyclic or Z(G) �Cm × H × Z

r in which Cm � Πp∈π(G/Z(G))Z(G)p, H � Πp�∈π(G/Z(G))Z(G)p andr ≥ 0 is the torsion-free rank of Z(G) and G/Z(G) has finite exponent.

Note that since gcd(|H |, |G/Z(G)|) = 1, we omit the condition “exp(G/

Z(G))|m”.

Corollary 2.4 ([4, Corollary 0.2]). Let G be a finitely generated group whichis not torsion-free. Then C∗ = Inn(G) if and only if G is nilpotent of class 2 andZ(G) � Cm×H×Z

r in which Cm � Πp∈π(G/Z(G))Z(G)p, H � Πp�∈π(G/Z(G))Z(G)p

and r ≥ 0 is the torsion-free rank of Z(G) and G/Z(G) has finite exponent.

Theorem 2.5 ([4, Theorem 2.1]). Let G be a finitely generated nilpotent groupof class 2. Then Hom( G

Z(G) , Z(G)) � GZ(G) if and only if Z(G) is infinite cyclic or

Z(G) � Cm×H×Zr in which Cm � Πp∈π(G/Z(G))Z(G)p, H � Πp�∈π(G/Z(G))Z(G)p

and r ≥ 0 is the torsion-free rank of Z(G) and G/Z(G) has finite exponent.

We also need the following replacements:

(i) In the abstract, the condition “Z(G) � Cm × Zr where G

Z(G) has expo-nent dividing m and r is the torsion-free rank of Z(G)” should be replacedby “Z(G) � Cm × H × Z

r in which Cm � Πp∈π(G/Z(G))Z(G)p, H �Πp�∈π(G/Z(G))Z(G)p and r ≥ 0 is the torsion-free rank of Z(G) and G/Z(G)has finite exponent”.

(ii) In Remark 2.5, “Z(G) � Z, Cm or Cm × Zr” should be replaced by “Z(G) is

infinite cyclic or Z(G) � Cm × H × Zr in which Cm = Πp∈π(G/Z(G))Z(G)p,

H � Πp�∈π(G/Z(G))Z(G)p and r ≥ 0 is the torsion-free rank of Z(G) andG/Z(G) has finite exponent”.

(iii) In p. 1286, in line 9 “the center of G is cyclic” should be replaced by “Z(G) �Cm × H in which Cm � Πp∈π(G/Z(G))Z(G)p and H � Πp�∈π(G/Z(G))Z(G)p”.

(iv) Finally in the proof of Theorem 2.1, we need to modify the proof as below:In Step 1, “Z(G) is cyclic” should be replaced by “Z(G) � Cm × H in whichCm = Πp∈π(G/Z(G))Z(G)p, H � Πp�∈π(G/Z(G))Z(G)p” and in Step 3, we con-sider exp(A) = n and “Let A = Cm × H . . .” should be replaced by “LetA � Cn × H where . . . Hom(C, A) � Hom(C, Cn) × Hom(C, H). It followsfrom Corollary 2.2 that C � C × Hom(C, H) × As · · · exp( G

Z(G) ) | exp(A) = n.Therefore C � Hom(C, A) and by Corollary 2.2, A � Cm × K whereCm � Πp∈π(G/Z(G))Ap and K � Πp�∈π(G/Z(G))Ap. Consequently, Z(G) �Cm ×K ×Z

r and . . . Z(G) � Cm ×K ×Zr where G

Z(G) has exponent dividing

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m, gcd(m, |H |) = 1 and . . .

Hom(

G

Z(G), Z(G)

)� Hom

(G

Z(G), Cm × K × Z

r

)

� Hom(

G

Z(G), Cm × K

)× Hom

(G

Z(G), Zr

)· · · ”.

Now, we turn to the proof of Corollary 2.1.

Proof of Corollary 2.1. First we prove that if G′ ≤ M then, Hom(G/Z(G), M) �G/Z(G) if and only if M � Cm × H × Z

r in which Cm � Πp∈π(G/Z(G))Mp, H �Πp�∈π(G/Z(G))Mp and r ≥ 0 is the torsion-free rank of M .

Let M = A × B where A = T (M) and B is torsion-free part of M . So B is afree abelian group of finite rank r ≥ 0, so

Hom(G/Z(G), M) � Hom(G/Z(G), A × Zr) � Hom(G/Z(G), A).

Since G is nilpotent group of class 2 and G/Z(G) is finite, by [4, Lemma 2.3],exp(G/Z(G)) = exp(G′) divides exp(T (M)), consequently by Corollary 2.2,Hom(G/Z(G), A) � G/Z(G) if and only if A = Cm × H where Cm �Πp∈π(G/Z(G))Ap and H � Πp�∈π(G/Z(G))Ap. Now by Theorem A(ii), AutM

N (G) =Inn(G) if and only if G′ ≤ M , N ≤ Z(G) and Hom(G/Z(G), M) � G/Z(G)or equivalently N = Z(G), G′ ≤ M and M � Cm × H × Z

r in which Cm �Πp∈π(G/Z(G))Mp, H � Πp�∈π(G/Z(G))Mp and r ≥ 0 is the torsion-free rank of M .

Let G be a finitely generated non-abelian group. We consider the case M =N = Z(G). If G/Z(G) is finite then by Corollary 2.1, Inn(G) = C∗ if and onlyif G is a nilpotent group of class 2 and Z(G) � Cm × H × Z

r in which Cm �Πp∈π(G/Z(G))Z(G)p, H � Πp�∈π(G/Z(G))Z(G)p and r ≥ 0 is the torsion-free rank ofZ(G). This result is also a consequence of Corollary 0.2 in [4].

In particular, if G is finite p-group, it is equivalent to Z(G) begin cyclic. There-fore, as a consequence of this result, we derive the main theorem of Shabani Attarin [2].

Corollary 2.6. Let G be a finite p-group. Then C∗ = Inn(G) if and only if G isabelian or a nilpotent group of class 2 and Z(G) is cyclic.

Since for any group G, Z2(G) ≤ CG(G′), the main result of Curran [5] follows.

Corollary 2.7. Let G be a finite non-abelian p-group. If Autc(G) = Z(Inn(G))then Z(G) ≤ G′. Also Autc(G) = Z(Inn(G)) if and only if Hom(G/G′, Z(G)) �Z2(G)/Z(G).

Proof. By taking M = Z(G) and N = G′ in Theorem A, it is sufficient toshow that Z(G) ≤ G′. On the contrary, if Z(G) ≤ G′ then G′ < G′Z(G)

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and G′ ∩ Z(G) < Z(G). So by [6, Lemma D], |Hom(G/G′Z(G), G′ ∩ Z(G))| <

|Hom(G/G′, Z(G))|. On the other hand since Autc(G) = C∗, by Lemma 1.4 wehave Autc(G) � Hom(G/G′, Z(G)). Now Z(Inn(G)) = C∗ implies that C∗ �Hom(G/G′Z(G), Z(G) ∩ G′). Consequently

|Hom(G/G′Z(G), G′ ∩ Z(G))| = |Hom(G/G′, Z(G))|.This contradiction completes the proof.

The proof of Z(G) ≤ G′ in Corollary 2.7 is similar to the proof in [5] by Curran.But here we prove it for convenience.

The following result of Curran and McCaughan [6] also follows.

Corollary 2.8. Let G be a finite non-abelian p-group. Then Autc(G) = Inn(G) ifand only if G′ = Z(G) and Z(G) is cyclic.

Another interesting result of Theorem A is indicated in the following corollary,that is Corollary 2.6 in [3].

Corollary 2.9. Let G be a finite non-abelian p-group. Then CAutc(G)(φ(G)) =Inn(G) if and only if φ(G) = Z(G) and Z(G) is cyclic.

Proof. First assume that CAutc(G)(φ(G)) = Inn(G). Hence G is a nilpotent groupof class 2 and by Theorem A(ii), φ(G) is central. On the other hand Z(G) ≤ φ(G),since CAutc(G)(φ(G)) ≤ Inn(G) (see [3, Proposition 2.2]). Consequently, φ(G) =Z(G) and hence C∗ = CAutc(G)(φ(G)) = Inn(G). Therefore by Corollary 2.6, Z(G)is cyclic.

Conversely, if φ(G) = Z(G) and Z(G) is cyclic then by using Corollary 2.6,CAutc(G)(φ(G)) = C∗ = Inn(G), as required.

Notice that this result can be generalized as follows.

Corollary 2.10. Let G be a finite non-abelian p-group and N be a normal subgroupof G such that Z(G) ≤ N . Then CAutc(G)(N) = Inn(G) if and only if N = Z(G)and Z(G) is cyclic.

Example 2.11. (i) Let G = 〈x, y : x8 = y2 = 1, xy = x−1〉. We have Z(G) =〈x4〉 � C2 and Z2(G) = 〈x2〉 � C4. Let N = 〈x2, y〉, then N is a normalsubgroup of G, containing Z(G). It is clear that Z2(G) ≤ CG(N). Thereforeby Theorem A(i), Z(Inn(G)) ≤ AutZ(G)

N (G) = CAutc(G)(N).(ii) Let G = 〈x, y : x8 = y2n

= 1, xy = x−1〉, n ≥ 2. We have G′ = 〈x2〉,Z(G) = 〈x4, y2〉 � C2 × C2n−1 and Z2(G) = 〈x2, y2〉 � C4 × C2n−1 . LetN = Z2(G), then Z(G) ≤ Z2(G) ≤ CG(Z2(G)) since Z2(G) is an abeliangroup. Therefore by Theorem A(i), Z(Inn(G)) ≤ CAutc(G)(Z2(G)). On theother hand CAutc(G)(Z2(G)) � C4

2 and Z2(G)/Z(G) � C2. Consequently byTheorem A(ii), Z(Inn(G)) < CAutc(G)(Z2(G)).

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(iii) Let G = 〈x, y, z : x2 = y8 = z2 = 1 = [x, z], yx = y5, yz = yx〉. Wehave Z(G) = 〈y4〉 � C2 and Z2(G) = 〈x, y2〉 � C2 × C4. Let N = Z2(G),then Z(G) ≤ Z2(G) ≤ CG(Z2(G)) since Z2(G) is an abelian group. There-fore by Theorem A(i), Z(Inn(G)) ≤ CAutc(G)(Z2(G)). On the other handCAutc(G)(Z2(G)) � Hom(G/Z2(G), Z(G)) � C2

2 and Z2(G)/Z(G) � C22 .

Therefore by Theorem A(ii), Z(Inn(G)) = CAutc(G)(Z2(G)).

Let G be a group and M1, M2, N1 and N2 be normal subgroups of G such thatM1 ≤ M2 and N2 ≤ N1 then clearly AutM1

N1(G) ≤ AutM2

N2(G). Now we continue by

proving Theorem B.

Proof of Theorem B. We proceed with a series of steps. Let G be a finite p-group.

Step 1. Let M be a central subgroup of exponent pn and N be a normal subgroupof G. Then

⋂f∈Hom(G/N,M) ker f = G′Gpn

N/N .

Proof. Clearly G′Gpn

N/N ≤ ker f for all f ∈ Hom(G/N, M). To prove the con-verse inclusion, let x ∈ G′Gpn

N . Note that since M ≤ Z(G), it follows thatHom(G/N, M) � Hom(G/G′N, M). Put G = G/G′N , G is a finite abelianp-group and so there exist x1, . . . , xt ∈ G such that G = 〈x1〉 × · · · × 〈xt〉and xG′N = xps1

1 · · ·xpst

t G′N for suitable si ≥ 0 (see [7, Lemma 2.2]). Sincex ∈ G′Gpn

N/G′N , xpsj

j ∈ Gpn

for some j and therefore sj < n. Now choose elementz ∈ M such that |z| = min{|xj |, pn}. And define a homomorphism fz : xj → z fromG to M . Recall that if M ≤ Z(G) and K is a direct factor of G then any element f

of Hom(K, M) induces an element f of Hom(G, M) which is trivial on the comple-ment of K in G. To simplify the notion, we will identify f with the correspondinghomomorphism from G to M . We have fz(x) = fz(xj

psj ) = zpsj = 1. Thereforex ∈ ⋂f∈Hom(G/N,M) ker f and consequently the equality holds.

Step 2. Let N1 and N2 be two normal subgroups of G such that N2 ≤ N1 and M ≤Z(G) ∩ Ni for i = 1, 2. Then AutM

N1(G) = AutM

N2(G) if and only if N1 ≤ G′Gpn

N2

where exp(M) = pn.

Proof. Since N2 ≤ N1, AutMN1

(G) ≤ AutMN2

(G). Suppose that N1 ≤ G′Gpn

N2

then by Step 1, N1 ≤ ker f for all f ∈ Hom(G/N2, M). Therefore we haveHom(G/N2, M) � Hom(G/N1N2, M) � Hom(G/N1, M), since N2 ≤ N1. Thatis |AutM

N1(G)| = |AutM

N2(G)| and hence AutM

N1(G) = AutM

N2(G).

Conversely, assume that AutMN1

(G) = AutMN2

(G) then α(n) = n for all n ∈ N1

and α ∈ AutMN2

(G). By Lemma 1.2, α∗(n) = 1 for all α∗ ∈ Hom(G/N2, M) andn ∈ N1. Consequently by Step 1, N1 ≤ G′Gpn

N2, as required.

Step 3. Let N be a normal subgroup of G and M1 < M2 ≤ Z(G) ∩ N . ThenAutM1

N (G) = AutM2N (G) if and only if s = s′ and exp(G/G′N) ≤ pat where t is the

smallest integer between 1 and s such that aj = bj for all t + 1 ≤ j ≤ s.

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Proof. Since M1 ≤ M2, AutM1N (G) ≤ AutM2

N (G) and so by Lemma 1.2, AutM1N (G) =

AutM2N (G) if and only if Hom(G/N, M1) � Hom(G/N, M2). First assume that

AutM1N (G) = AutM2

N (G), then clearly s = s′. So by applying Lemma 1.5 withA = M1, B = M2, and C = G/G′N we get exp(G/G′N) ≤ pat (for if not,exp(G/G′N) ≥ pat+1 then we have |Hom(G/N, M1)| < |Hom(G/N, M2)|, a contra-diction). Now suppose that s = s′ and exp(G/G′N) ≤ pat then |Hom(G/N, M1)| =|Hom(G/N, M2)| and therefore AutM1

N (G) = AutM2N (G).

Step 4. Conclusion.First assume that AutM1

N1(G) = AutM2

N2(G). By Lemma 1.2, Hom(G/N1, M1) �

Hom(G/N2, M2). If M1 < M2 and N2 < N1 then by Lemma D [6], Hom(G/

N1, M1) < Hom(G/N2, M2). By this contradiction we have M1 = M2 or N1 = N2.If M1 = M2 then by Step 2, N1 ≤ G′Gpn

N2. Else since M1 = M2, N1 = N2 thenby Step 3, it follows that s = s′ and exp(G/G′N1) ≤ pat .

Conversely, if (i) or (ii) holds then it is easy to see that Hom(G/N1, M1) �Hom(G/N2, M2). On the other hand, since M1 ≤ M2 and N2 ≤ N1, AutM1

N1(G) ≤

AutM2N2

(G) and consequently AutM1N1

(G) = AutM2N2

(G), as required.

Note that in the proof of Theorem B, we use the conditions Mi ≤ Z(G) ∩ Ni

only to prove the equality |AutMi

Ni(G)| = |Hom(G/Ni, Mi)|. So by Lemma 1.1 we

may substitute this condition by “G be a purely non-abelian group”. And by usingthe same argument, one can easily prove the following proposition.

Proposition 2.12. Let G be a finite purely non-abelian p-group. Let M1 and M2

be two central subgroups and N1 and N2 be two normal subgroups of G such thatM1 ≤ M2 and N2 ≤ N1. Then AutM1

N1(G) = AutM2

N2(G) if and only if one of the

following statements holds :

(i) M1 = M2 and N1 ≤ G′Gpn

N2 where exp(M1) = pn, or(ii) N1 = N2, s = s′ and exp(G/G′N1) ≤ pat where t is the smallest integer between

1 and s such that aj = bj for all t + 1 ≤ j ≤ s.

The Theorem B has a number of important consequences. As a first applicationof this we get the following result.

Let G be a finite p-group and M be a central subgroup of G. Then we maywrite

M = Cpa1 × Cpa2 × · · · × Cpas ,

Z(G) = Cpb1 × Cpb2 × · · · × Cpbs′ ,

where a1 ≥ a2 ≥ · · · ≥ as > 0 and b1 ≥ b2 ≥ · · · ≥ bs′ > 0 are cyclic decompositionsof M and Z(G). If t is the smallest integer between 1 and s such that aj = bj

for all t + 1 ≤ j ≤ s then by using the above notation we state the followingcorollary.

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Corollary 2.13. Let G be a finite p-group. Let M and N be two normal subgroupsof G such that M ≤ Z(G) ≤ N . Then


(a) M = Z(G) and N ≤ G′Gpn

Z(G) where exp(Z(G)) = pn, or(b) N = Z(G), s = s′ and exp(G/G′Z(G)) ≤ pat.

(ii) AutMN (G) = Autc(G) if and only if one of the following statements holds :

(c) M = Z(G) and N ≤ G′Gpn

where exp(Z(G)) = pn, or(d) N ≤ G′, s = s′ and exp(G/G′) ≤ pat .

Proof. (i) Follows immediately by applying Theorem B with M1 = M , N1 = N

and M2 = N2 = Z(G). To prove (ii), note that if M is a central subgroup thenAutM

N (G) = AutMNG′(G). Also from M ≤ Z(G) ≤ N , it follows that Autc(G) =

C∗ and by Lemma 1.4, G is a purely non-abelian group. Therefore by applyingProposition 2.12 with M1 = M , N1 = NG′, M2 = Z(G) and N2 = G′, (c) or (d)holds.

Conversely, if (c) holds then

Hom(G/N, M) � Hom(G/N, Z(G)) � Hom(G/G′N, Z(G))

� Hom(G/G′, Z(G)).

Now if (d) holds then

Hom(G/N, M) � Hom(G/G′N, M) � Hom(G/G′, M) � Hom(G/G′, Z(G)).

Hence |Hom(G/N, M)| = |Hom(G/G′, Z(G))| or equivalently AutMN (G) = Autc(G).

Notice that the Corollary 2.13 is a restatement of the Corollary C (parts (i)and (ii)).

This corollary yields the following corollary that is the main theorem of Jafari [7].

Corollary 2.14. Let G be a finite p-group. Then Autc(G) = C∗ if and only ifZ(G) ≤ G′Gpn

where exp(Z(G)) = pn.

Proof. It suffices from Corollary 2.13(ii) to assume M = N = Z(G).

Next is a special case of what happens in the group of nilpotency class 2. Weuse the same notations as Corollary C.

Corollary 2.15. Let G be a finite non-abelian p-group of class 2. Let M and N betwo normal subgroups of G such that M ≤ Z(G) ≤ N . Then


(a) M = N = Z(G), or(b) N = Z(G), s = s′ and exp(G/Z(G)) ≤ pat .

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(ii) AutMN (G) = Autc(G) if and only if M = N = Z(G) = G′Gpn

whereexp(Z(G)) = pn or equivalently M = N = Z(G), r = r′, (G/Z(G))/H �(G/G′)/K and exp(Z(G)) = exp(G′).

Proof. (i) First assume that AutMN (G) = C∗. By Corollary 2.13(i), one of the

following statements holds:

(a) M = Z(G) and N ≤ Gpn

Z(G). Since G is of class 2, Inn(G) ≤ AutMN (G) = C∗

and so N is central. Hence in this case M = N = Z(G). Otherwise(b) N = Z(G), s = s′ and exp(G/G′Z(G)) ≤ pat .

The converse follows immediately from Corollary 2.13(i).(ii) If AutM

N (G) = Autc(G) then AutMN (G) = C∗ = Autc(G). Since

C∗ = Autc(G), Corollary 2.14 implies that Z(G) ≤ G′Gpn

. On the otherhand exp(G/Z(G)) ≤ exp(Z(G)) = pn and hence Gpn ≤ Z(G). ConsequentlyZ(G) = G′Gpn

and since AutMN (G) = C∗, by (i) we have N = Z(G). Also

from C∗ = Autc(G), it follows that G is purely non-abelian and |Autc(G)| =|Hom(G/G′, Z(G))|. Now Inn(G) ≤ AutM

N (G) implies that G′ ≤ M . We claimM = Z(G). Otherwise if M < Z(G) then G′ < Z(G) so by Lemma D [6],|Hom(G/Z(G), M)| < |Hom(G/G′, Z(G))|. But AutM

Z(G)(G) = Autc(G) and by thiscontradiction M = Z(G), as required. Now suppose M = N = Z(G) = G′Gpn

thenC∗ = Autc(G) and Hom(G/Z(G), Z(G)) � Hom(G/G′, Z(G)). Therefore evidentlyr = r′ and by using Lemma 1.5 we may conclude that (G/Z(G))/H � (G/G′)/K

and exp(Z(G)) = exp(G′). Finally assume that M = N = Z(G), r = r′,(G/Z(G))/H � (G/G′)/K and exp(Z(G)) = exp(G′). Then by Lemma 1.5,Hom(G/Z(G), Z(G)) � Hom(G/G′, Z(G)). But |Hom(G/Z(G), Z(G))| = |C∗| ≤|Autc(G)| ≤ |Hom(G/G′, Z(G)). So AutM

N (G) = C∗ = Autc(G) as required.

Note that the Corollary 2.15 is a generalization of a result of Jafari in [7].Moreover, Corollary 2.15(ii) yields the following corollary that is the Corollary 2.7of Jafari in [7]. In particular, the fact that (i) is equivalent to (iii), establishes themain theorem of Yadav in [9]. We use the same notations as Corollary C.

Corollary 2.16. Let G be a finite non-abelian p-group of class 2. Then the follow-ing are equivalent :

(i) Autc(G) = C∗,(ii) Z(G) = G′Gpn

where exp(Z(G)) = pn,

(iii) r = r′, (G/Z(G))/H � (G/G′)/K and exp(Z(G)) = exp(G′).

Proof. It is sufficient in Corollary 2.15(ii), we assume M = N = Z(G).

Next we prove that the conditions “M1 ≤ M2 and N2 ≤ N1” can be removed.

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Let G be a finite p-group. Let M1 and M2 be two central subgroups of G. Thenwe may write

M1 ∩ M2 = Cpa1 × Cpa2 × · · · × Cpas ,

M1 = Cpb1 × Cpb2 × · · · × Cpbs1 ,

M2 = Cpc1 × Cpc2 × · · · × Cpcs2 ,

where a1 ≥ a2 ≥ · · · ≥ as > 0 and b1 ≥ b2 ≥ · · · ≥ bs1 > 0 and c1 ≥ c2 ≥ · · · ≥cs2 > 0. Let t1 be a smallest integer between 1 and s such that aj = bj for allt1 + 1 ≤ j ≤ s and t2 be a smallest integer between 1 and s such that aj = cj forall t2 + 1 ≤ j ≤ s. Write exp(Mi) = pni . Using the above notation we state thefollowing corollary.

Corollary 2.17. Let G be a finite p-group. Let M1, M2, N1 and N2 be normalsubgroups of G such that Mi ≤ Z(G)∩Ni for i = 1, 2. Then AutM1

N1(G) = AutM2

N2(G)

if and only if one of the following statements holds :

(i) M1 = M2 and Ni ≤ G′GpnjNj for i, j = 1, 2 and i = j,

(ii) M1 ≤ M2, s = s1 = s2, N1 ≤ N2 ≤ G′Gpn1N1 and exp(G/G′N2) ≤ pat2 ,

(iii) M2 ≤ M1, s = s1 = s2, N2 ≤ N1 ≤ G′Gpn2N2 and exp(G/G′N1) ≤ pat1 ,

(iv) N1 = N2, s = s1 = s2 and exp(G/G′N1) ≤ pati for i = 1, 2.

Proof. First assume that AutM1N1

(G) = AutM2N2

(G). Therefore we have AutM1N1

(G) =AutM1∩M2

N1N2(G) = AutM2

N2(G). Clearly, M1 ∩ M2 ≤ Mi and Ni ≤ N1N2 for i = 1, 2

and so we may apply Theorem B. Since AutMi

Ni(G) = AutM1∩M2

N1N2(G), for i = 1, 2

one of the following case happens:

(I) Mi = M1 ∩ M2 and N1N2 ≤ G′GpniNi. So Mi ≤ Mj and Nj ≤ G′Gpni

Ni fori = j. Or,

(II) Ni = N1N2, s = si and exp(G/G′Ni) ≤ pati . So Nj ≤ Ni, s = si andexp(G/G′Ni) ≤ pati for i = j.

Therefore we have the following four cases:

(1) If for i = 1, 2, (I) holds, then M1 = M2 and Ni ≤ G′GpnjNj for i, j = 1, 2 and

i = j and hence (i) follows.(2) If for i = 1, (I) and for i = 2, (II) happen, then M1 ≤ M2, N2 ≤ G′Gpn1

N1 andalso s = s2, N1 ≤ N2 and exp(G/G′N2) ≤ pat2 . Since at2 ≤ n1, Gpn1 ≤ G′N2

and N1 ≤ N2 implies that G′N2 = G′Gpn1N1. Furthermore from M1 ≤ M2,

it follows that s = s1 and consequently, M1 ≤ M2, s = s1 = s2, N1 ≤ N2 ≤G′Gpn1

N1 and exp(G/G′N2) ≤ pat2 and so in this case (ii) holds.(3) If for i = 1, (II) and for i = 2, (I) happen, then with an argument similar to

the case (2) we may conclude that (iii) holds.(4) Finally if for i = 1, 2, (II) holds, then evidently N1 = N2 = N1N2, s = s1 = s2

and exp(G/G′N1) ≤ pati where i = 1, 2, that is (iv).

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Conversely, First assume that (i) holds. Then we have M1 = M2 = M andfrom Ni ≤ G′Gpnj

Nj , it follows that Hom(G/Nj , M) � Hom(G/N1N2, M) fori, j = 1, 2 and i = j. Consequently AutM

Ni(G) = AutM

N1N2(G) and hence AutM

N1(G) =

AutMN2

(G).Now suppose that (ii) holds. Since N2 = N1N2, s = s2 and exp(G/G′N2) ≤ pat2

we have AutM1∩M2N1N2

(G) = AutM2N1N2

(G) = AutM2N2

(G). Also N2 ≤ G′Gpn1N1 con-

cludes that AutM1N1

(G) = AutM1N1N2

(G) = AutM1∩M2N1N2

(G) since M1 ≤ M2. ThereforeAutM1

N1(G) = AutM2

N2(G). The case (iii) follows by a similar argument.

Finally suppose that (iv) holds. So N1 = N2 = N , s = s1 = s2 andexp(G/G′Ni) ≤ pati for i = 1, 2, it follows that AutMi

N (G) = AutM1∩M2N (G) and

this completes the proof.

Note that here also, the condition “Mi ≤ Z(G)∩Ni for i = 1, 2” can be replacedby condition “G be a purely non-abelian group”.

Corollary 2.18. Let G be a finite purely non-abelian p-group. Let M1, M2, N1

and N2 be normal subgroups of G such that Mi are central for i = 1, 2. ThenAutM1

N1(G) = AutM2

N2(G) if and only if one of the following statements holds :

(i) M1 = M2 and Ni ≤ G′GpnjNj for i, j = 1, 2 and i = j,

(ii) M1 ≤ M2, s = s1 = s2, N1 ≤ N2 ≤ G′Gpn1N1 and exp(G/G′N2) ≤ pat2 ,

(iii) M2 ≤ M1, s = s1 = s2, N2 ≤ N1 ≤ G′Gpn2N2 and exp(G/G′N1) ≤ pat1 ,

(iv) N1 = N2, s = s1 = s2 and exp(G/G′N1) ≤ pati where i = 1, 2.

Another interesting equality is indicated by the following result.

Proposition 2.19. Let G be a finite purely non-abelian p-group. Let M, N1 andN2 be normal subgroups of G such that M is central. If the invariants of M (in thecyclic decomposition) are greater than or equal to exp(G/G′Ni) for i = 1, 2 thenAutM

N1(G) = AutM

N2(G) if and only if G′N1 = G′N2.

Proof. Let M = Cpa1 × Cpa2 × · · · × Cpas and exp(G/G′Ni) = pni for i = 1, 2.First assume that N2 ≤ N1. By assumption aj ≥ ni for all 1 ≤ j ≤ s and i = 1, 2.Consequently for i = 1, 2 we have

Hom(G/Ni, M) � Hom(G/Ni, Cpa1 ) × · · · × Hom(G/Ni, Cpas )

� (G/G′Ni)n.

Therefore AutMN1

(G) = AutMN2

(G) if and only if G/G′N1 = G/G′N2 or equivalentlyG′N1 = G′N2. Since AutM

N1(G) = AutM

N2(G) if and only if AutM

Ni(G) = AutM

N1N2(G)

for i = 1, 2 the general case follows.

Acknowledgment

I thank Professor S. K. Sehgal the editor of the Journal of Algebra and Its Appli-cations and the referee who have patiently read and verified this note.

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References

[1] J. E. Adney and T. Yen, Automorphisms of a p-group, Illinois J. Math. 9 (1965)137–143.

[2] M. Shabani Attar, On central automorphisms that fix the center elementwise, Arch.Math. 89 (2007) 296–297.

[3] M. Shabani Attar, On a conjecture about automorphisms of finite p-groups, Arch.Math. 93 (2009) 399–403.

[4] Z. Azhdari and M. Akhavan-Malayeri, On inner automorphisms and central automor-phisms of nilpotent group of class 2, J. Algebra Appl. 10(4) (2011) 1283–1290.

[5] M. J. Curran, Finite groups with central automorphism group of minimal order, Math.Proc. R. Ir. Acad. A 104(2) (2004) 223–229.

[6] M. J. Curran and D. J. McCaughan, Central automorphisms that are almost inner,Commun. Algebra 29(5) (2001) 2081–2087.

[7] S. H. Jafari, Central automorphism groups fixing the center element-wise, Int. Elec-tron. J. Algebra 9 (2011) 167–170.

[8] O. Muller, On p-automorphisms of finitep-groups, Arch. Math. 32(1) (1979) 533–538.[9] M. K. Yadav, On central automorphisms fixing the center element-wise, Commun.

Algebra 37 (2009) 4325–4331.

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on automorphisms fixing certain groups

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