Finite groups in which every self-centralizing subgroup is a TI-subgroup or subnormal or has $p'$-order

TL;DR

This paper characterizes the structure of finite groups where every self-centralizing subgroup is either a TI-subgroup, subnormal, or has order coprime to a fixed prime, extending to all subgroups under these conditions.

Contribution

It provides complete characterizations of finite groups with specific subgroup properties and establishes equivalences between properties of all subgroups and just self-centralizing subgroups.

Findings

Characterization of finite groups with all subgroups being TI, subnormal, or p'-order

Equivalence between properties of all subgroups and self-centralizing subgroups

Structural descriptions of such finite groups

Abstract

We first give complete characterizations of the structure of finite group $G$ in which every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) is a TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$. Furthermore, we prove that every self-centralizing subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$ if and only if every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or subnormal or has $p'$-order. Based on these results, we obtain the structure of finite group $G$ in which every self-centralizing subgroup (or non-nilpotent subgroup, or non-abelian subgroup) is a TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$.