• Kyungpook Mathematical Journal
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• v.59 no.2
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• pp.203-207
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• 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.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.121-123
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• 2016

In this paper, we compute the number of distinct centralizers and commutativity degree of a class of finite groups. These computations produce a further class of examples of groups answering one question raised by Belcastro and Sherman and another one raised by Lescot.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.513-522
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• 2012

In this paper, we prove common fixed point theorems for families of occasionally weakly compatible mappings in Menger spaces using implicit relation. Our results extend and generalize the results of Altun and Turkoglu [9] in the sense that the concept of occasionally weakly compatible maps is the most general among all the commutativity concepts. Also the completeness of the whole space, continuity of the involved maps and containment of ranges amongst involved maps are completely relaxed.

• Journal of the Korean Mathematical Society
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• v.49 no.4
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• pp.855-865
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• 2012

Recently, we have introduced a group invariant, which is related to the number of elements $x$ and $y$ of a finite group $G$ such that $x{\wedge}y=1_{G{\wedge}G}$ in the exterior square $G{\wedge}G$ of $G$. This number gives restrictions on the Schur multiplier of $G$ and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form $h^m{\wedge}k$ of $H{\wedge}K$ such that $h^m{\wedge}k=1_{H{\wedge}K}$, where $m{\geq}1$ and $H$ and $K$ are arbitrary subgroups of $G$.

• Journal of the Korean Mathematical Society
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• v.50 no.1
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• pp.161-171
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• 2013

Given an integer $n{\geq}1$ and a compact group G, we find some restrictions for the probability that two randomly picked elements $x^n$ and $y$ of G commute. In the case $n=1$ this notion was investigated by W. H. Gustafson in 1973 and its influence on the structure of the group has been studied in the researches of several authors in last years.