KYUNGPOOK Math. J. 2019; 59(2): 203-207  
The Commutativity Degree in the Class of Nonabelian Groups of Same Order
Hojjat Rostami
Department of Education, Molla Sadra Teaching and Training Research Center, Zanjan 13858899, Iran
h.rostami5991@gmail.com
Received: July 30, 2017; Revised: March 29, 2019; Accepted: April 23, 2019; Published online: June 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. In this paper we give a sharp upper bound of commutativity degree of nonabelial groups in terms of their order.

Keywords: finite group, commutativity degree, nonnilpotent.
1. Introduction

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. In the other words, the commutativity degree of a finite group G is the ratio

cp(G)={(x,y)G×G:[x,y]=1}G2.

It is easy to see that cp(G)=xGCG(x)G2 where CG(x) is the centralizer of x in G. Also G is abelian if and only if cp(G) = 1.

During the last few decades, there has been a growing interest in the study of finite groups in terms of their commutativity degree. This ratio has been investigated by many authors. For example, Gustafson in [4] showed cp(G)58 for all non-abelian groups G. Lescot, in [6], determined all groups G with cp(G)12. Guranlik and G. R. Robinson considered all groups with cp(G)>340 and proved that such groups are either solvable or are isomorphic to A5×C2n where n ≥ 1 (see [3]). Another results about cp(G) can be found in [1] and [9].

Our result is as follows.

Theorem 1.1

Let G be a non-abelian group of ordern=p1n1p2n2prnrwhere ni >0 for each i and p1, ··· , pr are distinct primes. Then

  • (1) cp(G)p2+p-1p3and the equality holds if and only ifGZ(G)Zp×Zpwhere p is the smallest prime divisor ofGZ(G).

  • (2) Let G be a non-nilpotent group. Thencp(G)max{piti+pj2-1pitipj2:         pjpiti-1,1i,jr,1tini}.

    and the equality holds if and only ifGZ(G)Zpiti×Zpjfor integers i and j such thatpjpiti-1and 1 ≤ tini.

First part of above theorem generalizes Lemma 1.3 of [5] and the second part generalizes a result of Lescot which can be found in [5].

Throughout the paper all groups are finite and p is a prime. Also, for gG, we often use to denote the coset gZ(G), and use for the factor group HZ(G)Z(G) where H is any subgroup of G. Other notation is standard and can be found in [8].

2. Results

A minimal non-abelin group is a non-abelian group all of whose subgroups are abelian. Following lemma gives an important property of minimal non-abelian p-groups. It will be used in the proof of Lemma 2.2.

Lemma 2.1

Let G be a minimal non-abelian p- group. ThenGZ(G)Zp×Zp.

Proof

Since G is non-abelian, there exist two elements x and y such that xyyx. Therefore CG(x) and CG(y) are two distinct centralizers of G. By hypothesis these centralizers are abelian and have index p. On the other hand if t ∈ (CG(x) ∩ CG(y)) Z(G) then CG(t) = CG(x) = CG(y). This is a contradiction. Thus CG(x) ∩ CG(y) = Z(G) and so GZ(G)Zp×Zp as a desired.

Using the above lemma and Theorem 10.1.7 of [8] and Exercise 9.1.11 of [8], we can say that every non-abelian group contains either a minimal non-abelian p-subgroup or a subgroup with Frobenius central factor. The following lemma play an important role in the proof of our main theorem.

Lemma 2.2

Let G be a group. If G is non-abelian, then G contains a subgroup H such that eitherHZ(H)ZrtZqor Zp × Zp where p, q and r are distinct primes.

Proof

Towards contradiction, let G be a counter-example of minimal order. If H is a non-abelian subgroup of G, then it, and consequently G, has a subgroup with the desired structure. So we may assume that G is a minimal non-abelian group. Using Theorem 10.1.7 of [8] and Exercise 9.1.11 of [8] and Lemma 2.1 we get a contradiction.

The following lemma, which will be used in prove of Theorem 1.1, gives some relations between commutativity degree of certain groups and their sections and subgroups.

Lemma 2.3

Let G be a finite group.

  • (1) For every proper subgroup H of G, we have cp(G) ≤ cp(H).

  • (2) Whenever NG, we havecp(G)cp(GN).

  • (3) For every section, X, of G, we have cp(G) ≤ cp(X).

Proof

See proof of Lemma 2 of [3].

Lemma 2.4

Let G be a non-abelian group.

  • (1) IfGZ(G)ZptZq, thencp(G)=pt+q2-1ptq.

  • (2) IfGZ(G)Zp×Zp, thencp(G)=p2+p-1p3.

Proof

Let G be a non abelian group and GZ(G)ZptZq. Since HZ(H) has an abelian subgroup of index q, Theorem A of [2] tells us that is a Frobenius group. Now, all centralizers of are either of order q or pt. By counting all of them and using the definition we get the result in first case.

Now let G be a group with central factor isomorphic to Zp × Zp. It is clear that all centralizers of noncentral elements of G are abelian of index p. So as a result we have

cp(G)=(G-Z(G)(Gp))+G2p2G2=p2+p-1p3.

Now we recall a nilpotent number defined in [7].

Definition 2.5

A positive integer n is called a nilpotent number if every group of order n is nilpotent.

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1

It is clear that there is at least one non-abelian group G of order n such that p is the smallest prime divisor of GZ(G). We denote some p-sylow subgroup of G by P. First we suppose that P is non-abelian and thus all Sylow p-subgroups P of G are non-abelian. By Lemma 2.2, P contains some minimal non abelian subgroup. Therefore by Lemma 2.3 and Lemma 2.4 we have the result in first case. Now let cp(G)=p2+p-1p3 but GZ(G) be not isomorphic to Zp × Zp. Then by Lemmas 2.2, 2.3 and 2.4 we have a contradiction. Now if GZ(G)Zp×Zp, then Lemma 2.4 gives the result in first case.

Now let G be a non-nilpotent group. Thus n is not a nilpotent number and so there are positive integers r and s such that r|si − 1 for some integer i. Also let

ϒ={(r,s)rsare primes,rsi-1but rsj-1for all integers j<i}.

Put also Γ={rt+s2-1rts2(r,s)ϒ}. We know by [7] that ϒ and so Γ are non-empty. Now we choose (p, q) ∈ ϒ such that qα+p2-1qαp2 is a maximum element of Γ. We now define a group H.

It is well known that Aut(Zqi)PGLi(q) and so

Aut(Zqi)=(qi-q)(qi-q)(qi-qi-1).

Therefore there is some θAut(Zqi) with |θ| = q. Let

ι:ZpAut(Zqi)

be a group homomorphism such that ι(a) = θ and Zp = 〈a〉. Now ZqiZp is a semidirect group with respect to ι and we denote ZqiZp×A by H in which A is abelian group with A=nqip. By Lemma 2.4, we have cp(H)=qi+p2-1qip.

Now we prove that for all non-nilpotent groups G of order n, cp(G)pi+q2-1piq. If all proper subgroups of G are nilpotent, then using Theorem 10.1.7 of [8] and Exercise 9.1.11 of [8] one can see that GZ(G) is Frobenius group whose Frobenius kernel is elementary abelian and whose Frobenius complement is of prime order. Now assume that G contains at least one proper non-nilpotent subgroup. Therefore by Lemma 2.2, G has a proper subgroup whose central factor is a Frobenius group with elementary abelian Frobenius kernel and cyclic Frobenius complement of prime order. Anyway, we can assume that K is a subgroup of G (perhaps G itself ) such that KZ(K)ZrtZs where r, s are distinct primes and t is an positive integer. By Lemma 2.3, cp(G)cp(KZ(K))=rt+s2-1rts2. But by choosing (p, q) ∈ ϒ, we have cp(G) ≤ cp(H) and proof is complete.

In the above theorem, if |G| is even, then we have the following.

Corollary 2.6

Let G be a non- nilpotent group of order n. Thencp(G)p+34pand equality holds if and only ifGZ(G)D2p.

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