Resumen de Groups with specific number of centralizers

A. Abdollahi, S. M. Jafarian Amiri, Mohammadi Hassanabadi

• 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 |cent(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.