Finite groups determined by the number of element centralizers

Abstract

ABSTRACTFor a finite group G, let |Cent(G)| and ω(G) denote the number of centralizers of its elements and the maximum size of a set of pairwise noncommuting elements of it, respectively. A group G is called n-centralizer if |Cent(G)| = n and primitive n-centralizer if . In this paper, among other results, we find |Cent(G)| and ω(G) when is minimal nonabelian and this generalizes some previous results. We give a necessary and sufficient condition for a primitive n-centralizer group G with the minimal nonabelian central factor. Also we show that if , then G is a primitive 14-centralizer group and ω(G) = 10 or 13. Finally we confirm Conjecture 2.4 in [A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7(2) (2000), 139-146].