Characterizations of Some Groups in Terms of Centralizers

A group G is said to be n-centralizer if its number of element centralizers $$\mid {{\,\mathrm{Cent}\,}}(G)\mid =n$$ , an F-group every non-central centralizer contains no other and a CA-group all are abelian. For any non-abelian G, we prove that \frac{G}{Z(G)}\mid \le (n-2)^2$$ $$n 12$$ 2(n-4)^{{log}_2^{(n-4)}}$$ otherwise, which improves earlier result. We arbitrary F-group, then gcd $$(n-2, \mid ) \ne 1$$ . finite show \ge \frac{\mid }{2}$$ iff $$G \cong A_4 $$ extraspecial 2-group or Frobenius with abelian kernel complement order 2. Among results, for non-trivial center, it proved = 2-group. give family F-groups not CA-groups extend