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
Department of Education, Molla Sadra Teaching and Training Research Center, Zanjan 13858899, Iran
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.
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
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  showed
Our result is as follows.
and the equality holds if and only if where p is the smallest prime divisor of .
Let G be a non-nilpotent group. Then and the equality holds if and only if for integers i and j such that and1 ≤ t i≤ n i .
Throughout the paper all groups are finite and
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
Using the above lemma and Theorem 10.1.7 of  and Exercise 9.1.11 of , we can say that every non-abelian group contains either a minimal non-abelian
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.
Let G be a finite group.
For every proper subgroup H of G, we have cp( G) ≤ cp( H).
Whenever N⊴ G, we have .
For every section, X, of G, we have cp( G) ≤ cp( X).
See proof of Lemma 2 of .
Let G be a non-abelian group.
If , then .
If , then .
Now we recall a nilpotent number defined in .
A positive integer
Now we are ready to prove Theorem 1.1.
It is clear that there is at least one non-abelian group
It is well known that
Therefore there is some
be a group homomorphism such that
Now we prove that for all non-nilpotent groups
In the above theorem, if |
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