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ON NONCOMMUTING SETS AND CENTRALISERS IN INFINITE GROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 1 / February 2016
- Published online by Cambridge University Press:
- 08 July 2015, pp. 42-46
- Print publication:
- February 2016
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- Article
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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.
