### Article

KYUNGPOOK Math. J. 2019; 59(2): 203-207

**Published online** June 23, 2019

Copyright © Kyungpook Mathematical Journal.

### 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

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

It is easy to see that _{G}

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

Our result is as follows.

### Theorem 1.1

_{i}_{1}, ··· , _{r}

(1)

$cp(G)\le {\scriptstyle \frac{{p}^{2}+p-1}{{p}^{3}}}$ and the equality holds if and only if ${\scriptstyle \frac{G}{Z(G)}}\cong {Z}_{p}\times {Z}_{p}$ where p is the smallest prime divisor of $\mid {\scriptstyle \frac{G}{Z(G)}}\mid $ .(2)

Let G be a non-nilpotent group. Then $$cp(G)\le max\{\frac{{p}_{i}^{{t}_{i}}+{p}_{j}^{2}-1}{{p}_{i}^{{t}_{i}}{p}_{j}^{2}}:\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{p}_{j}\mid {p}_{i}^{{t}_{i}}-1,1\le i,j\le r,1\le {t}_{i}\le {n}_{i}\}.$$ and the equality holds if and only if ${\scriptstyle \frac{G}{Z(G)}}\cong {Z}_{{p}_{i}}^{{t}_{i}}\times {Z}_{{p}_{j}}$ for integers i and j such that ${p}_{j}\mid {p}_{i}^{{t}_{i}}-1$ and 1 ≤t ≤_{i}n _{i}.

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

### 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

### Lemma 2.1

**Proof**

Since _{G}_{G}_{G}_{G}_{G}_{G}_{G}_{G}_{G}

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

### Lemma 2.2

_{p}_{p}

**Proof**

Towards contradiction, let

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 N ⊴G, we have $cp(G)\le cp({\scriptstyle \frac{G}{N}})$ .(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)

If ${\scriptstyle \frac{G}{Z(G)}}\cong {Z}_{p}^{t}\u22ca{Z}_{q}$ ,then $cp(G)={\scriptstyle \frac{{p}^{t}+{q}^{2}-1}{{p}^{t}q}}$ .(2)

If ${\scriptstyle \frac{G}{Z(G)}}\cong {Z}_{p}\times {Z}_{p}$ ,then $cp(G)={\scriptstyle \frac{{p}^{2}+p-1}{{p}^{3}}}$ .

**Proof**

Let ^{t}

Now let _{p}_{p}

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

### Definition 2.5

A positive integer

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 _{p}_{p}

Now let ^{i}

Put also

It is well known that

Therefore there is some

be a group homomorphism such that _{p}

Now we prove that for all non-nilpotent groups

In the above theorem, if |

### Corollary 2.6

### References

- AK. Das, and RK. Nath.
A characterisation of certain finite groups of odd order . Math Proc R Ir Acad.,111A (2)(2011), 69-78. - S. Dolfi, M. Herzog, and E. Jabara.
Finite groups whose noncentral commuting elements have centralizers of equal size . Bull Aust Math Soc.,82 (2010), 293-304. - RM. Guralnick, and GR. Robinson.
On the commuting probability in finite groups . J Algebra.,300 (2)(2006), 509-528. - WH. Gustafson.
What is the probability that two group elements commute? . Amer Math Monthly.,80 (9)(1973), 1031-1034. - P. Lescot.
Isoclinism classes and commutativity degrees of finite groups . J Algebra.,177 (3)(1995), 847-869. - P. Lescot.
Central extensions and commutativity degree . Comm Algebra.,29 (10)(2001), 4451-4460. - J. Pakianathan, and K. Shankar.
Nilpotent numbers . Amer Math Monthly.,107 (7)(2000), 631-634. - DJS. Robinson. A course in the theory of groups,
, Springer-Verlag, New York, 1996. - DJ. Rusin.
What is the probability that two elements of a finite group commute? . Pacific J Math.,82 (1)(1979), 237-247.