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Criteria for Commutativity in Large Groups

Published online by Cambridge University Press:  20 November 2018

A. Mohammadi Hassanabadi  and
Akbar Rhemtulla
A. Mohammadi Hassanabadi
Affiliation:
Department of Mathematics University of Isfahan Isfahan Iran
Akbar Rhemtulla
Affiliation:
Department of Mathematical Sciences University of Alberta Edmonton, Alberta T6G 2G1
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Abstract

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In this paper we prove the following:

  1. 1. Let $m\ge 2,\,n\ge 1$ be integers and let $G$ be a group such that ${{(XY)}^{n}}\,=\,{{(YX)}^{n}}$ for all subsets $X,Y$ of size $m$ in $G$. Then

    1. a) $G$ is abelian or a $\text{BFC}$-group of finite exponent bounded by a function of $m$ and $n$.

    2. b) If $m\ge n$ then $G$ is abelian or $|G|$ is bounded by a function of $m$ and $n$.

  2. 2. The only non-abelian group $G$ such that ${{(XY)}^{2}}\,=\,{{(YX)}^{2}}$ for all subsets $X,Y$ of size 2 in $G$ is the quaternion group of order 8.

  3. 3. Let $m$, $n$ be positive integers and $G$ a group such that

    $${{X}_{1}}\cdot \cdot \cdot \,{{X}_{n}}\,\subseteq \,\bigcup\limits_{\sigma \in {{S}_{n}}\,\backslash \,1}{{{X}_{\sigma (1)}}\cdot \cdot \cdot \,{{X}_{\sigma (n)}}}$$

for all subsets ${{X}_{i}}$ of size $m$ in $G$. Then $G$ is $n$-permutable or $|G|$ is bounded by a function of $m$ and $n$.

Keywords

20E3420F24
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 1 , 01 March 1998 , pp. 65 - 70
Copyright
Copyright © Canadian Mathematical Society 1998

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