On Noncommuting Sets and Centralisers in Infinite Groups

Abstract

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.

Groups with Exactly Ten Centralizers

“…All n-centralizer groups have been characterized for n ≤ 9 (see [1,2,5] and [9]). M. Zarrin [22] have generalized the previous results for infinite groups. He [19] showed that the derived length of a solvable group G is ≤ |Cent(G)|.…”

Groups with the same number of centralizers

“…It is easy to see that two simple groups are isomorphic if and only if they are isoclonic. Zarrin in [10] proved that for every two isoclinic groups G and S, |cent(G)| = |cent(S)|. The natural question is whether the converse of his statement is true?…”

On the norm of the centralizers of a group

“…(See Theorem 2.2 of [6] and also Theorem B of [7].) This result suggests that the behavior of centralizers has a strong influence on the structure of the group (for more information see [8] and [9]). This is the main motivation to introduce a new series of norms in groups by their normalizers of the centralizers.…”