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′ 4 Ω1(G) ≤ Z(G) respectively, if p is odd, and that G ≤ G and Ω2(G) ≤ Z(G) respectively, if p = 2. Now a new question may be formulated: which other finite non-cyclic p-groups G with |G| ≥ p3 have |G| dividing |Aut(G)|? In this short note, we prove that for G a finite non-cyclic p-group with |G| ≥ p3, if G has an abelian maximal subgroup, or if G has elementary abelian centre and CG(Z(Φ(G))) 6= Φ(G), then |G| divides |Aut(G)|. The latter is a partial generalization of Gasch¨utz’ result [6] that |G| divides |Aut(G)| when the centre has order p.

Notation. We use standard notation in group theory. All groups are assumed to be finite and p always stands for a prime number. For M,N normal M subgroups in G, we set AutN (G) to be the subgroup of automorphisms of G M that centralize G/M and N, and let OutN (G) be its corresponding image in Out(G). When M or N is Z(G), we write just Z for conciseness. On the other hand, if G is a finite p-group then Ω1(G) denotes the subgroup generated by all elements of G of order p.

2. An abelian maximal subgroup Let G be a finite p-group with an abelian maximal subgroup A. We collect here a few well-known results (see [9, Lemma 4.6 and its proof]). Theorem 2.1. Let G be a group having an abelian normal subgroup A such that the quotient G/A = hgAi, with g ∈ G, is cyclic. Then (i) G′ = {[a, g] | a ∈ A} and (ii) |G′| = |A : A ∩ Z(G)|. Corollary 2.2. Let G be a finite non-abelian p-group having an abelian maximal subgroup. Then |G : Z(G)| = p|G′|. In [13], Webb uses the following approach to find non-inner automor- phisms of p-power order, which we will use in the forthcoming theorem. For a maximal subgroup M of G, we first define two homomorphisms on Z(M). Let g ∈ G be such that G/M = hgMi, then − p−1 τ : m 7→ g p(gm)p = mg +...+g+1 − γ : m 7→ [m, g]= mg 1 for all m ∈ Z(M). We have im γ ⊆ ker τ and im τ ⊆ ker γ = Z(G) ∩ M. Corollary 2.3. [13] Let G be a finite non-abelian p-group and M a maximal subgroup of G containing Z(G). Then G has a non-inner automorphism of p-power order inducing the identity on G/M and M if and only if im τ 6= ker γ. We remark that the proof of the above also tells us that if im τ 6= ker γ, M then |OutM (G)| = | ker γ|/|im τ|. A NOTE ON AUTOMORPHISMS OF FINITE p-GROUPS 3

Theorem 2.4. Let G be a finite non-cyclic p-group with |G|≥ p3 and with an abelian maximal subgroup A. Then |G| divides |Aut(G)|. Proof. We work with the subgroup AutZ(G) of central automorphisms in Aut(G). Now Z Z Z |Aut (G)| · |Inn(G)| |Aut (G)| · |G/Z(G)| |Aut (G)Inn(G)| = Z = |Aut (G) ∩ Inn(G)| |Z2(G)/Z(G)| Z and hence it suffices to show that |Aut (G)| ≥ |Z2(G)|. According to Otto [11], when G is the direct product of an abelian p- group H and a p-group K having no abelian direct factor, then one has |H| · |Aut(K)|p divides |Aut(G)|. Hence, we may assume that G has no abelian direct factor. It then follows by Adney and Yen [1] that |AutZ (G)| = |Hom(G/G′,Z(G))|. By Corollary 2.2, we have |G : Z(G)| = p|G′| and equally, ′ (1) |G : G | = p|Z(G)|. Let H = G/Z(G). Then A/Z(G) is an abelian maximal subgroup of H. Applying Corollary 2.2 to H yields ′ |H : H | = p|Z(H)|, so ′ |G : G Z(G)| = p|Z2(G) : Z(G)|. Hence 1 ′ ′ (2) |Z2(G)| = · |G ∩ Z(G)| · |G : G |. p Combining this with (1) gives ′ (3) |Z2(G)| = |G ∩ Z(G)| · |Z(G)|. Next we have G′ = {[a, g] | a ∈ A} by Theorem 2.1. By Webb’s construc- tion, we know that im γ ⊆ ker τ and here im γ = [A, g]= G′. Now for a ∈ A and g as in Theorem 2.1, we have p−1 [a, g]g +...+g+1 = 1. We claim that exp(G′ ∩ Z(G)) = p. For, if [a, g] ∈ Z(G), then [a, g]g = [a, g]. Consequently, p−1 1 = [a, g]g +...+g+1 = [a, g]p and thus o([a, g]) ≤ p. Clearly the minimal number d := d(G) of generators of G is at least 2. In order to proceed, we divide into the following two cases: (a) exp(G/G′) ≥ exp Z(G), and (b) exp(G/G′) ≤ exp Z(G). Case (a): Suppose exp(G/G′) ≥ exp Z(G). We express G/G′ as ′ ′ ′ ′ G/G = hg1G i × hg2G i× . . . × hgdG i ′ ′ where o(g1G ) = exp(G/G ) and by assumption d ≥ 2. 4 G.A.FERNANDEZ-ALCOBER´ AND A. THILLAISUNDARAM

′ ′ We consider homomorphisms from G/G to Z(G). The element g1G may ′ be mapped to any element of Z(G), and g2G may be mapped to any element of G′ ∩ Z(G), which is of exponent p. Thus, with the aid of (3), ′ ′ |Hom(G/G ,Z(G))| ≥ |Z(G)| · |G ∩ Z(G)| = |Z2(G)|. Case (b): Suppose exp(G/G′) ≤ exp Z(G). Similarly we express Z(G) as Z(G)= hz1i × hz2i× . . . × hzri where r = d(Z(G)) and o(z1) = exp Z(G). We consider two families of homomorphisms from G/G′ to Z(G). First, ′ G/G → Z(G)

′ bi giG 7→ z1 bi ′ for 1 ≤ i ≤ d, where bi is such that o(z1 ) divides o(giG ). This gives rise to |G/G′| homomorphisms. Next, we consider all homomorphisms from G/G′ to Z(G) where each ′ element giG is mapped to any element of order p in hz2i× . . . × hzri. This gives ′ − − |Ω1(Z(G))| |G ∩ Z(G)| (pr 1)d ≥ pr 1 = ≥ p p different homomorphisms. Multiplying both together and then using (2), we obtain ′ ′ ′ |G/G | · |G ∩ Z(G)| |Hom(G/G ,Z(G))|≥ = |Z2(G)|. p 

3. Elementary abelian centre Let G be a finite p-group with elementary abelian centre. In order to prove that |G| divides |Aut(G)|, we may assume, upon consultation of [5] and the final remarks in [8], that Z(G) < Φ(G). One of the following three cases exclusively occurs. Case 1. Z(M)= Z(G) for some maximal subgroup M of G. Case 2. Z(M) ⊃ Z(G) for all maximal subgroups M of G. Then either (A) CG(Z(Φ(G))) 6= Φ(G); or (B) CG(Z(Φ(G))) = Φ(G).

The main result of this section is to show that if G is a finite p-group with elementary abelian centre and not in Case 2B, then |G| divides |Aut(G)|. With regard to Case 2B, we would also like to mention another long- standing conjecture for finite p-groups: does there always exist a non-inner automorphism of order p? The case 2B is the only remaining case for this conjecture (see [3]). First we deal with Case 1. A NOTE ON AUTOMORPHISMS OF FINITE p-GROUPS 5

Lemma 3.1. [10, Lemma 2.1(b)] Suppose M is a maximal subgroup of G. Z If Z(M) ⊆ Z(G) then AutM (G) is a non-trivial elementary abelian p-group Z such that AutM (G) ∩ Inn(G) = 1. We comment that the proof of this result in [10] tells us that Z |AutM (G)| = |Hom(G/M, Z(M))| = |Ω1(Z(M))|. Lemma 3.2. Let G be a finite p-group with elementary abelian centre. Sup- pose that Z(M) = Z(G) for some maximal subgroup M of G. Then |G| divides |Aut(G)|. Proof. Using Lemma 3.1 and the above comment, it follows that Z |Out(G)|p ≥ |AutM (G)| = |Z(G)|. Hence |G| divides |Aut(G)| as required.  Next, suppose G is as in Case 2(A). We will need the following. Theorem 3.3. [13] Let G be a finite non-abelian p-group. Then p divides the order of OutZ (G). Now we present our result. Proposition 3.4. Let G be a finite p-group with elementary abelian cen- tre, such that CG(Z(Φ(G))) 6= Φ(G) and Z(M) ⊃ Z(G) for all maximal subgroups M of G. Then |G| divides |Aut(G)|. Proof. By M¨uller [10, proof of Lemma 2.2], there exist maximal subgroups M and N such that G = Z(M)N and Z(G) = Z(M) ∩ N. Write G/M = hgMi for some g ∈ G, and consider the homomorphism

τM : Z(M) −→ Z(M) p−1 m 7−→ g−p(gm)p = mg +···+g+1. Since Z(M) Z(M) Z(M)N G = =∼ = Z(G) Z(M) ∩ N N N and Z(G) ⊆ ker τM , it follows that |im τM | ≤ p. If im τM = 1, the remark M after Corollary 2.3 implies that |OutM (G)| = |Z(G)| and we are done. Hence we assume that im τM =∼ Cp. If Z(G) =∼ Cp, then |G| divides |Aut(G)| by Gasch¨utz [6]. So we may assume that |Z(G)| > p. Again by the same M remark, we have |OutM (G)| = |Z(G)|/p. By [10, Lemma 2.2], it follows that G is a central product of subgroups R and S, where R/Z(R) =∼ Cp × Cp and Z(R) = Z(G) = Z(S) = R ∩ S. Furthermore, R = Z(M)Z(N) and S = M ∩ N = CG(R). By Theorem 3.3, there exists a non-inner automorphism β ∈ AutZ (S) of p-power order. As observed by M¨uller [10, Section 3], the automorphism β extends uniquely to some non-inner γ ∈ AutZ (G) with trivial action on R. Certainly γ does not act trivially on M ∩ N = S, so γ 6∈ AutM (G). Let M γ be the image of γ in Out(G). We now show that γ 6∈ OutM (G). On the 6 G.A.FERNANDEZ-ALCOBER´ AND A. THILLAISUNDARAM

M contrary, suppose that γ = ρ.ι where ρ ∈ AutM (G) and ι ∈ Inn(G). As S ≤ M, we have β(s) = γ(s) = ρ(s)x = sx for all s ∈ S and a fixed x ∈ G. ′ ′ Writing x = rs for some r ∈ R, s ∈ S and recalling that S = CG(R), we ′ have β(s)= ss . This implies that β ∈ Inn(S), a contradiction. Thus M |Out(G)|p ≥ |hγ, OutM (G)i| ≥ |Z(G)|. It follows that |G| divides |Aut(G)|.  The authors are grateful to the various people who helped to read and improve this manuscript.

References [1] J. E. Adney and T. Yen, Automorphisms of a p-group, Ill. J. Math. 9 (1965), 137–143. [2] M. Couson, On the character degrees and automorphism groups of finite p-groups by coclass, PhD Thesis, Technische Universit¨at Braunschweig, Germany, 2013. [3] M. Deaconescu and G. Silberberg, Noninner automorphisms of order p of finite p- groups, J. Algebra 250 (2002), 283-287. [4] B. Eick, Automorphism groups of 2-groups, J. Algebra 300 (1) (2006), 91–101. [5] R. Faudree, A note on the automorphism group of a p-group, Proc. Amer. Math. Soc. 19 (1968), 1379–1382. [6] W. Gasch¨utz, Nichtabelsche p-Gruppen besitzen ¨aussere p-Automorphismen (Ger- man), J. Algebra 4 (1966), 1–2. [7] J. Gonz´alez-S´anchez and A. Jaikin-Zaipirain, Finite p-groups with small automor- phism group, Forum Math. Sigma 3 (2015), e7. [8] K. G. Hummel, The order of the automorphism group of a central product, Proc. Amer. Math. Soc. 47 (1975), 37–40. [9] I. M. Isaacs, Finite group theory, Graduate Studies in Mathematics, Vol. 92, Amer. Math. Soc., Providence, 2008. [10] O. M¨uller, On p-automorphisms of finite p-groups, Arch. Math. 32 (1979), 553–538. [11] A. D. Otto, Central automorphisms of a finite p-group, Trans. Amer. Math. Soc. 125 (1966), 280–287. [12] A. Thillaisundaram, The automorphism group for p-central p-groups, Internat. J. Group Theory 1 (2) (2012), 59–71. [13] U. H. M. Webb, An elementary proof of Gasch¨utz’ theorem, Arch. Math. 35 (1980), 23–26. [14] M. K. Yadav, On automorphisms of finite p-groups, J. Group Theory 10 (6) (2007), 859–866.

Gustavo A. Fernandez-Alcober:´ Department of Mathematics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain E-mail address: [email protected]

Anitha Thillaisundaram: Mathematisches Institut, Heinrich-Heine- Universitat,¨ 40225 Dusseldorf,¨ Germany E-mail address: [email protected]