Characterizations of some groups in terms of centralizers
From MaRDI portal
Publication:6378242
DOI10.1007/S00025-022-01687-4zbMath1510.20022arXiv2109.10530MaRDI QIDQ6378242
Publication date: 22 September 2021
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 $mid frac{G}{Z(G)}mid leq (n-2)^2$, if $n leq 12$ and $mid frac{G}{Z(G)}mid leq 2(n-4)^{{log}_2^{(n-4)}}$ otherwise, which improves an earlier result. We prove that if $G$ is an arbitrary non-abelian $n$-centralizer F-group, then gcd$(n-2, mid frac{G}{Z(G)}mid)
eq 1$. 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.
This page was built for publication: Characterizations of some groups in terms of centralizers
