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 ncentralizer
group if #Cent(G)=n, and a primitive ncentralizer 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
6centralizer 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.
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