On Finite Groups with a Given Number of Centralizers

Abstract

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n ≠ 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) ≅ A4, the alternating group on four letters. Also, we prove that, if G/Z(G) ≅ A4, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) ≅ A4 and #Cent(G) = 8.