ON FINITE GROUPS WITH A CERTAIN NUMBER OF CENTRALIZERS

Let G be a finite group and #Cent(G) denote the number of centralizers of its elements. G is called n-centralizer if #Cent( G) = n, and primitive n-centralizer if #Cent( G) = #Cent(G/Z(G)) = n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and if G is a finite group such that G/Z(G) similar or equal to A(5), then #Cent(G) = 22 or 32. Moreover, we prove that A(5) is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A(5) in terms of the number of centralizers