  Let $G$ be a finite non-abelian group and $m=|G|/|Z(G)|$. In this paper we
investigate $m$-centralizer group $G$ with cyclic center and we will prove that
if $G$ is a finite non-abelian $m$-centralizer $CA$-group, then there exists an
integer $r>1$ such that $m=2^r.$ It is also prove that if $G$ is an
$m$-centralizer non-abelian finite group which is not a $CA$-group and its
derived subgroup $G'$ is of order 2, then there exists an integer $s>1$ such
that $m=2^{2s}.$
