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.