On the maximum number of the pairwise noncommuting elements in a finite group

摘要

For a finite group G, let omega(G) be the maximum size of a set of pairwise noncommuting elements in G. In this paper, we give an upper bound of omega(G) for an arbitrary nilpotent group G. As an application of this result, we give a partial answer to Question 2.8 of [A.R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7(2) (2000) 139-146]. Also we compute omega(G) when G/Z(G) is a Frobenius group. Finally we describe structural properties of all groups G with omega(G) - 5.