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ON NONCOMMUTING SETS AND CENTRALISERS IN INFINITE GROUPS

Part of: Special aspects of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  08 July 2015

MOHAMMAD ZARRIN
MOHAMMAD ZARRIN*
Affiliation:
Department of Mathematics, University of Kurdistan, PO Box 416, Sanandaj, Iran email m.zarrin@uok.ac.ir, zarrin@ipm.ir
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Abstract

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A subset $X$ of a group $G$ is a set of pairwise noncommuting elements if $ab\neq ba$ for any two distinct elements $a$ and $b$ in $X$. If $|X|\geq |Y|$ for any other set of pairwise noncommuting elements $Y$ in $G$, then $X$ is called a maximal subset of pairwise noncommuting elements and the cardinality of such a subset (if it exists) is denoted by ${\it\omega}(G)$. In this paper, among other things, we prove that, for each positive integer $n$, there are only finitely many groups $G$, up to isoclinism, with ${\it\omega}(G)=n$, and we obtain similar results for groups with exactly $n$ centralisers.

Keywords

pairwise noncommuting elements of a groupn-centralisers

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems
Secondary: 20F99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 1 , February 2016 , pp. 42 - 46
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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