Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-24T01:54:27.119Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Blyth, R. D., Rewriting products of group elements, I. J. Algebra 116 (1988), 506–521.Google Scholar
2. Blyth, R. D., Rewriting products of group elements, II. J. Algebra 116 (1988), 246–259.Google Scholar
3. Curzio, M., Longobardi, P., Maj, M. and Robinson, D. J. S., A permutational property of groups. Arch.Math. (Basel) 44 (1985), 385–389.Google Scholar
4. Lennox, John C., Hassanabadi, A. Mohammadi and Wiegold, James, Some commutativity criteria, II. Rend. Sem. Mat. Univ. Padova. 86 (1991), 207–211.Google Scholar
5. Longobardi, P., Maj, M. and Stonehewer, S. E., The classification of groups in which every product of four elements can be reordered. Rend. Sem. Mat. Univ. Padova. 93 (1995), 7–26.Google Scholar
6. Maj, M. and Stonehewer, S. E., Non-nilpotent groups in which every product of four elements can be reordered. Canad. J. Math. 42 (1990), 1053–1066.Google Scholar
7. Neumann, P. M. and Vaughan-Lee, M. R., An essay on BFC-groups. Proc. London. Math. Soc. (3) 35 (1977), 213–237.Google Scholar