On SS-quasinormalities of the maximal subgroup series of finite groups

TL;DR

This paper studies a special class of subgroup series in finite groups called SS-quasinormal series, proving that their existence implies the group is solvable and characterizing supersolvability through subnormal SS-quasinormal series.

Contribution

It introduces the concept of SS-quasinormal maximal subgroup series and establishes their implications for the solvability and supersolvability of finite groups.

Findings

Groups with SS-quasinormal maximal subgroup series are solvable.

Supersolvability is characterized by the existence of a subnormal SS-quasinormal maximal subgroup series.

The paper provides necessary and sufficient conditions linking subgroup series properties to group structure.

Abstract

Let $G$ be finite group. A subgroup $H$ of $G$ is said to be an $SS$-quasinormal subgroup of $G$, if there exists a subgroup $B$ of $G$ such that $G = HB$ and $H$ permutes with every Sylow subgroup of $B$. Let $\Omega: G=G_0>G_1>\cdots>G_{n-1}>G_n=1$ be a maximal subgroup series of $G$, where $G_i$ is a maximal subgroup of $G_{i-1}$ for every $i = 1, \ldots , n$. In this paper, we investigate the finite groups $G$ that admit an $SS$-quasinormal maximal subgroup series, i.e., all $G_i$ are $SS$-quasinormal in $G$. First, we prove that if $G$ possesses an $SS$-quasinormal maximal subgroup series, then $G$ is solvable. Furthermore, we show that $G$ is supersolvable if and only if $G$ possesses an $SS$-quasinormal maximal subgroup series which is subnormal in $G$.