Groups with specific number of centralizers

Abstract

Let G be a. group and let cent(G) denote the set of centralizers of single elements of G. A group G is called n-centralizer if |eerat(G)| = n. In this paper, for a finite group G, we give some interesting relations between \cent(G)\ and the maximum number of the pairwise non-commuting elements in G. Also we characterize all n-centralizer finite groups for n = 7 and 8. Using these results we prove that there is no finite group G with the property that \cent(G)\ = \cent(G/Z(G))\ = 8, where Z(G) denotes the centre of G. This latter result answers positively a conjecture posed by A. R. Ashrafi. © 2007 University of Houston.