Characterizations of Some Groups in Terms of Centralizers

Abstract

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