On Finite Groups with a Given Number of Centralizers

For a nite group G ,l et Cent(G) denote the set of centralizers of single elements of G and #Cent(G )= jCent(G)j. 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 nite groups G and prove that, for any positive integer n 6 ; 3, there exists a nite group G with #Cent(G )= n, which is a question raised by Belcastro and Sherman (2). We investigate the structure of nite 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 ,t hen #Cent(G )=6o r 8, and construct a group G with G=Z(G) = A4 and #Cent(G )=8 . 2000 Mathematics Subject Classication: 20D99, 20E07