On Finite Groups with a Given Number of Centralizers

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 ) ≅ A 4 , the alternating group on four letters. Also, we prove that, if G/Z ( G ) ≅ A 4 , then #Cent( G ) = 6 or 8, and construct a group G with G/Z ( G ) ≅ A 4 and #Cent( G ) = 8.