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For a finite group $G$, let Cent$(G)$ denote the set of
centralizers of single elements of $G$ and
$\#$Cent$(G)=|\mathrm{Cent}(G)|$. $G$ 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 $n\neq 2,3$, 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 ${G}/{Z(G)}\cong A_4$, the alternating
group on four letters. Also, we prove that, if ${G}/{Z(G)}\cong
A_4$, then $\#$Cent$(G)=6$ or $8$, and construct a group $G$ with
${G}/{Z(G)}\cong A_4$ and $\#$Cent$(G)=8$.
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