Characterizations of some groups in terms of centralizers

Abstract

A group $G$ is said to be $n$-centralizer if its number of element centralizers $\mid \Cent(G)\mid=n$, 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 $\mid \Cent(G)\mid \geq \frac{\mid G \mid}{2}$ 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 $\mid \Cent(G)\mid = \frac{\mid G \mid }{2}$ iff $G$ is an extraspecial $2$-group. We give a family of F-groups which are not CA-groups and extend an earlier result.