THE AUTOMORPHISM GROUP OF A FINITE MINIMAL NON-ABELIAN p-GROUP

NON-ABELIAN p-GROUP

S. FOULADI and R. ORFI

Communicated by the former editorial board

LetGbe a finite minimal non-abelianp-group. In this paper we give a structure theorem for the central automorphism group ofG. Also forp= 2, a structure theorem is given for the full automorphism group ofG.

AMS 2010 Subject Classification: 20D15, 20D45.

Key words: automorphism group, minimal non-abelian p-group, central auto- morphisms

1. INTRODUCTION

Determining the order and the structure of the automorphism group of a finite p-group is an important problem in group theory. There have been a number of studies of the automorphism group ofp-groups. Most of them deal with the order of Aut(G), the automorphism group of G, see for example [1]

and [6]. Moreover various attempts have been made to find a structure for the automorphism group of a finite p-group, see [3] and [5].

In this paper we study the automorphism group of a finite minimal non- abelian p-group. A minimal non-abelian group is a non-abelian group such that all its proper subgroups are abelian. A presentation of these groups is given in ([2], §1, Exercise 8a). Let Gbe a finite minimal non-abelianp-group.

Some results about the order of Aut(G) is given in [6] when p > 2. In this paper first we find the order of Aut_c(G), the central automorphism group of G, and then we give a structure theorem for Aut_c(G). Moreover forp= 2 we find the order of Aut(G) and we show that Aut2(G), the 2-Sylow subgroup of Aut(G), is a split extension of Aut_c(G) by Z2, where Z2 is the cyclic group of order 2. Finally we give a structure theorem for Aut(G) when p = 2 (see Theorem 3.7). In particular we study a problem posed by Y. Berkovich ([2], Problem 700).

Throughout this paper, the following notation is used. pdenotes a prime number. We used(G) for the minimal number of generators of a groupG. The

MATH. REPORTS16(66),1(2014), 133–139

center ofGis denoted byZ =Z(G) and the Frattini subgroup ofGis denoted by Φ = Φ(G). If α is an automorphism of a group G and x is an element of G, we writex^α for the image ofxunderα. We write Autc(G) for the group of central automorphisms ofGand Aut_p(G) for thep-Sylow subgroup of Aut(G).

Semi direct product of two subgroups H and K of a group G is denoted by HnK when K is normal in G. Moreover we write H.K when H and K are subgroups of a group G and H∩K = 1. A non-abelian group that has no non-trivial abelian direct factor is said to be purely non-abelian. Also Zn is the cyclic group of order nand Z^k_n is the direct product ofk copies of Zn. All unexplained notation is standard and follows of [2].

2. SOME BASIC RESULTS

In this section we give some basic results that are needed for the main results of the paper. First we state the following theorem which give a presen- tation for minimal non-abelian p-groups.

Theorem2.1 ([2],§1, Exercise 8a). LetGbe a finite minimal non-abelian p-group. Then |G⁰| = p and G/G⁰ is abelian of rank 2 and G is one of the following groups:

(i) G1 =ha, b|a^pm =b^pn = 1, a^b =a^1+pm−1i, m≥2, n≥1 and |G₁|=p^m+n (G₁ is metacyclic).

(ii) G₂ = ha, b|a^pm = b^pn = c^p = 1,[a, b] = c,[a, c] = [b, c] = 1i is non- metacyclic of order p^m+n+1 and if p= 2, thenm+n >2. Next G⁰₂ is a maximal cyclic subgroup of G2. In this case we may assume thatm≥n.

(iii) G3 ∼=Q8.

Moreover Z(G_i) = Φ(G_i) for 1≤i≤3.

For the rest of the paper we use the notation of Theorem 2.1 and all groups are assumed to be finite.

Lemma 2.2. Let Gbe a minimal non-abelian p-group. Then (i) G is purely non-abelian,

(ii) Z(G1) =ha^pi × hb^pi and Z(G2) =ha^pi × hb^pi × hci, (iii) G₁/G⁰₁ =Zp^m−1×Zpⁿ and G₂/G⁰₂ =Zp^m×Zpⁿ.

Proof. (i) By the way of contradiction if G = A×B, where A is a non-trivial abelian subgroup of G and B is purely non-abelian, then Φ(G) = Φ(A)×Φ(B). This implies that |B/Φ(B)|=psinced(G) = 2, a contradiction.

(ii) and (iii) are evident.

Corollary 2.3. For a minimal non-abelian p-group, we have

(i) |Aut_c(G₁)|=p^3m+n−4 when n≥m and

|Aut_c(G₁)|=p^m+3n−3 when m > n, (ii) |Aut_c(G₂)|=p^m+3n−1 when m > n and

|Aut_c(G2)|=p^4m−2 when m=n, (ii) Aut_c(G₃)∼=Z²₂.

Proof. This is clear by using ([1], Theorem 1) and Lemma 2.2.

3. Aut(G)WHEN p= 2

LetG be a minimal non-abelian 2-group. In this section we give a struc- ture theorem for Aut(G).

Lemma 3.1. Any automorphism of G1 has the following form: a7→ aⁱb^j and b7→a^rb^s, where i, j, r and s satisfy the following conditions.

(i) If n ≥ m , then 0 ≤ i, r < 2^m, 0 ≤ j, s < 2ⁿ, (i,2) = (s,2) = 1 and 2^n−m+1|j.

(ii) If m > nand m >2, then 0≤i, r <2^m, 0≤j, s <2ⁿ, (i,2) = (s,2) = 1 and 2^m−n|r.

(iii) If (m, n) = (2,1), then G₁∼= Aut(G₁)∼=D₈.

Proof. (i) and (ii) are evident by using (Substitution Test) ([4], Proposi- tion 3, p. 44), Lemma 2.2(ii) and the fact thatG₁/Φ(G₁) =hΦaⁱb^j,Φa^rb^si.

(iii) This is obvious.

Corollary 3.2. The order ofAut(G₁) is obtained as follows:

|Aut(G₁)|=

(2^3m+n−3 n≥m

2^m+3n−2 m > n

Proof. This is an immediate consequence of the above lemma.

Lemma 3.3. We have

(i) Aut_c(G₁)∼= ((Z²₂×Z2^m−2 ×Z2ⁿ⁻²)n Z2^m−1).Z2^m−1 when n≥m >2.

(ii) Aut_c(G₁)∼=Z2ⁿ⁻² ×Z⁴2 when n≥m and m= 2.

(iii) Autc(G1)∼= ((Z²₂×Z2^m−2 ×Z2ⁿ⁻²)n Z2ⁿ⁻¹).Z2ⁿ when m > n >2.

(iv) Autc(G1)∼=Z2^m−2 ×Z²₂ when m > n and n= 1.

(v) Aut_c(G₁)∼= (Z₂^m−2×Z³₂).Z4 when m > n and n= 2.

Proof. (i) We define the maps α, β, γ, δ,α⁰ and β⁰ by a^α =a⁵,b^α = b, a^β = a, b^β = b⁵, a^γ = ab^2n−m+1, b^γ = b, a^δ = a, b^δ = a²b, a^α0 = a⁻¹, b^α0 = b, a^β0 = a and b^β0 = b⁻¹. Obviously α, β, γ, δ, α⁰ and β⁰ are central automorphisms of G₁ and are of orders 2^m−2, 2ⁿ⁻², 2^m−1, 2^m−1, 2 and 2 respectively. Moreover we see that H=hα, β, α⁰, β⁰i=hαi × hβi × hα⁰i × hβ⁰i

and H ≤ N_Autc(G1)(hγi). Also H ∩ hγi = 1 and hδi ∩Hhγi = 1 by Lemma 2.2(ii). Now we can complete the proof by using Corollary 2.3(i).

(ii) We define β,α⁰,β⁰,γ andδ as in (i) and we see thathβ, α⁰, β⁰, γ, δi= hβi × hα⁰i × hβ⁰i × hγi × hδi,as desired.

(iii) We define α,β, α⁰ and β⁰ as in (i) and define γ and δ by a^γ =ab², b^γ =b,a^δ=aand b^δ=ba^2m−n. This is easy to check that γ and δ are central automorphisms ofG1 with|γ|= 2ⁿ⁻¹ and|δ|= 2ⁿ. Also these automorphisms satisfy the same results as (i).

(iv) We defineα,α⁰andδas in (iii). Obviouslyhα, α⁰, δi=hαi×hα⁰i×hδi.

(v) We defineα,α⁰,β⁰,γ andδas in (iii) and we see thatK=hα, α⁰, β⁰, γi

=hαi × hα⁰i × hβ⁰i × hγi, and hδi ∩K= 1, as desired.

Now, we prove the same results for G2.

Lemma3.4. Any automorphism ofG₂has the following form: a7→aⁱb^jc^t, b7→a^rb^sc^t0 and c7→c, where i, j, t, r, s, andt⁰ satisfy the following conditions.

(i) If m > n, then0≤i, r <2^m, 0≤j, s <2ⁿ, 0≤t, t⁰<2,(i,2) = (s,2) = 1 and 2^m−n|r.

(ii) For m=n, if iis even, then r and j are odd. If i is odd and j is even, thens is odd. If iand j are odd, then only one of r or s is even.

Proof. This is clear.

Corollary 3.5. The order ofAut(G2) is obtained as follows:

|Aut(G₂)|=

(2^m+3n m > n 3.2^4m−1 m=n Lemma 3.6. We have

(i) Autc(G2)∼= ((Z²₂×Z2^m−2×Z2ⁿ⁻²)n Z2ⁿ⁻¹).(Z2ⁿ×Z²2) whenm > n≥2.

(ii) Autc(G2)∼=Z2^m−2 ×Z⁴₂ when m > n and n= 1.

(iii) Aut_c(G₂)∼= ((Z²₂×Z²₂m−2)n Z2^m−1).(Z2^m−1 ×Z²₂) when m=n >2 . (iv) Aut_c(G₂)∼=Z⁶₂ when (m, n) = (2,2).

Proof. (i) We define the maps α, β, γ, δ,α⁰ and β⁰ by a^α =a⁵,b^α = b, a^β = a, b^β = b⁵, a^γ =ab², b^γ = b, a^δ = a, b^δ = ba^2m−n, a^α0 =a⁻¹, b^α0 = b, a^β0 = a and b^β0 = b⁻¹. Obviously α, β, γ, δ, α⁰ and β⁰ are central auto- morphisms of G2 and are of orders 2^m−2, 2ⁿ⁻², 2ⁿ⁻¹, 2ⁿ, 2 and 2 respec- tively. Moreover we see that H =hα, β, α⁰, β⁰i =hαi × hβi × hα⁰i × hβ⁰i and H ≤ N_Autc(G2)(hγi). On setting K = hInn(G₂), δi, we have Hhγi ∩K = 1, Inn(G2)∩ hδi = 1 and H ∩ hγi = 1 by Lemma 2.2(ii). Also K is abelian since Aut_c(G₂) =C_Aut(G2)(Inn(G₂)). Now we can complete the proof by using Corollary 2.3(ii).

(ii) We define α, α⁰ and δ as in (i) and we see that hα, α⁰, δ,Inn(G₂)i= hαi × hα⁰i × hδi ×Inn(G₂),as desired.

(iii) We define α, β, α⁰, β⁰ and γ as in (i) and define δ by a^δ = a and b^δ =ba². This is easy to check that δ ∈Aut_c(G₂) and |δ|= 2^m−1. Also these automorphisms satisfy the same results as (i).

(iv) This is evident by using GAP [7].

Theorem 3.7. For the groupG1, we have Aut(G1) = Aut2(G1) = Z2n Aut_c(G₁) when (m, n) 6= (2,1) and Aut(G₁) ∼= D₈ when (m, n) = (2,1). For the group G2 we have Aut2(G2) = Z2 nAutc(G2). Moreover Aut(G2) = Aut2(G2) when m > n and Aut(G2) = Aut2(G2).Z3 when m = n. Also Aut(G₃) =S₄.

Proof. First we consider the groupG1 and we see that the mapψdefined bya^ψ =a⁻¹andb^ψ =bais a non-central automorphism of order 2 whenn≥m.

Also the mapψdefined bya^ψ =abandb^ψ =b⁻¹is a non-central automorphism of order 2 when m > n and m > 2. Therefore by corollaries 2.3 and 3.2, we see that Aut(G1) = Aut2(G1) =Z2nAutc(G1) when (m, n)6= (2,1). Also for (m, n) = (2,1), we have Aut(G₁) ∼= D₈. Now, for the group G₂, the map ψ defined bya^ψ =abandb^ψ =b⁻¹ is a non-central automorphism of order 2 and so Aut2(G2) =Z2nAutc(G2) by corollaries 2.3 and 3.5. Moreover the mapφ defined by a^φ=b⁻¹a⁻¹ and b^φ=ais an automorphism of order 3 inG₂ when m=n, completing the proof. Finally Aut(G₃) =S₄ by using GAP [7].

4. Aut(G)WHEN p >2

In this section, we give a structure theorem for Autc(G) when G is a minimal non-abelian p-group with p >2. First we give the following theorem from [6] for the order of Aut(G).

Theorem4.1. LetGbe a minimal non-abelianp-group withp >2. Then (i) |Aut(G₁)|= (p−1)p^3m+n−3 when n≥m and

|Aut(G₁)|= (p−1)p^3n+m−2 when m > n, (ii) |Aut(G₂)|= (p−1)²p^3n+m whenm > n and

|Aut(G₂)|= (p²−1)(p−1)p^4m−1 when m=n.

Proof. This follows from the presentations A.1 and B.1 in [6].

Corollary 4.2. Let G be a minimal non-abelian p-group with p > 2.

Then |Aut(G) : Aut_c(G)|=p.

Proof. This is obvious by Theorem 4.1 and Corollary 2.3.

Lemma 4.3. We have

(i) Aut_c(G₁)∼= ((Z_p^m−1 ×Zpⁿ⁻¹)n Zpⁿ⁻¹).Zpⁿ when m > n >1.

(ii) Autc(G1)∼=Zp^m−1 ×Zp when m > n and n= 1.

(iii) Autc(G1)∼= ((Zp^m−1 ×Zpⁿ⁻¹)n Zp^m−1).Zp^m−1 when n≥m.

Proof. (i) We define the maps α, β, γ and δ by a^α = a^1+p, b^α = b, a^β = a, b^β = b^1+p, a^γ = ab^p, b^γ = b, a^δ = a and b^δ = ba^pm−n. Obviously α, β, γ and δ, are central automorphisms of G1 and of orders p^m−1, pⁿ⁻¹, pⁿ⁻¹ and pⁿ respectively. Moreover we see that H =hα, βi =hαi × hβi and H ≤ N_Autc(G1)(hγi). AlsoH∩ hγi= 1 and hδi ∩Hhγi= 1 by Lemma 2.2(ii).

Now we can complete the proof by using Corollary 2.3(i).

(ii) We define α and δ as in (i) and we see that hα, δi=hαi × hδi as desired.

(iii) We define α and β as in (i) and define γ and δ by a^γ =ab^pn−m+1, b^γ = b, a^δ = a and b^δ = a^pb. This is easy to check that γ and δ are central automorphisms ofG₁ with|γ|=|δ|=p^m−1. Also these automorphisms satisfy the same results as (i).

Lemma 4.4. We have

(i) Aut_c(G₂)∼= ((Z_p^m−1 ×Zpⁿ⁻¹)n Zpⁿ⁻¹).(Zpⁿ×Z²p) when m > n >1.

(ii) Autc(G2)∼=Zp^m−1 ×Z³p when m > n and n= 1.

(iii) Autc(G2)∼= (Z²_pm−1n Zp^m−1).(Z_p^m−1 ×Z²_p) whenm=n >1.

(iv) Autc(G2)∼=Z²_p when m=n= 1.

Proof. (i) We define the maps α, β, γ and δ by a^α = a^1+p, b^α = b, a^β = a, b^β = b^1+p, a^γ = ab^p, b^γ = b, a^δ = a and b^δ = ba^pm−n. Obviously α,β, γ and δ are central automorphisms of G2 and are of orders p^m−1, pⁿ⁻¹, pⁿ⁻¹ and pⁿ respectively. Moreover we see that H =hα, βi =hαi × hβi and H ≤ N_Autc(G2)(hγi). On setting K = hInn(G₂), δi we have Hhγi ∩K = 1, Inn(G2)∩ hδi = 1 and H∩ hγi = 1 by Lemma 2.2(ii). Now we can complete the proof by using Corollary 2.3(ii).

(ii) We define α and δ as in (i) and we see thathα, δ,Inn(G₂)i =hαi × hδi ×Inn(G2), as desired.

(iii) We define α, β and γ as in (i) and define δ by a^δ = a and b^δ = ba^p. This is easy to check that δ ∈ Aut_c(G₂) and |δ| = p^m−1. Also these automorphisms satisfy the same results as (i).

(iv) By using Corollary 2.3(ii) and Lemma 2.2(ii), we see that Autc(G2) = Inn(G₂).

REFERENCES

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

[2] Y. Berkovich,Groups of Prime Power Order. Vol. 1, Walter de Gruyter, Berlin, 2008.

[3] S. Fouladi and R. Orfi,Automorphisms of metabelian prime power order groups of maxi- mal class. Bull. Aust. Math. Soc.77(2008), 261–276.

[4] D.L. Johnson,Presentation of groups. 2nd ed., LMS Stud. texts, Camberidge University press 1997.

[5] A. Juh´asz,The group of automorphisms of a class of finitep-groups. Trans. Amer. Math.

Soc. Vol. 270(1982),2, 469–481.

[6] F. Menegazzo,Automorphisms ofp-groups with cyclic commutator subgroup. Rend. Sem.

Mat. Univ. Padova, 90(1993), 81–101.

[7] TheGAPGroup,GAP- -Groups, Algorithms, and Programming. Version4.4.10; 2007.

(http://www.gap-system.org).

Received 8 December 2011 Kharazmi University,

Faculty of Mathematical Sciences and Computer, 50 Taleghani Ave.,

Tehran 1561836314, Iran s fouladi@khu.ac.ir Kharazmi University,

Faculty of Mathematical Sciences and Computer, 50 Taleghani Ave.,

Tehran 1561836314, Iran r orfi@tmu.ac.ir