A subset X of a group G is a set of pairwise noncommuting elements if ab , ba for any two distinct elements a and b in X. If |X| ≥ | 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 ω ( 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 n, and we obtain similar results for groups with exactly n centralisers.
