On strong $p$-$CAP$-subgroups of finite groups and saturated fusion systems

Shengmin Zhang, Zhencai Shen

Published 2023-12-27, updated 2024-02-06Version 2

A subgroup $A$ of a finite group $G$ is said to be a strong $p$-$CAP$-subgroup of $G$, if for any $A \leq H \leq G$, $A$ is a $p$-$CAP$-subgroup of $H$. A subgroup $A$ of a finite group $G$ is called a $p$-$CAP$-subgroup of $G$, if $A$ covers or avoids any $pd$-chief factor of $G$. Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that $S_{i+1}/S_i$ is cyclic, $i=0,1,\cdots, n-1$, $S_i$ is strongly $\mathcal{F}$-closed, $i=0,1,\cdots,n$. In this paper, we investigate the properties of strong $p$-$CAP$-subgroups and give several characterizations for a group $G$ to be $p$-supersoluble under the hypothesis that some subgroups of $G$ are strong $p$-$CAP$-subgroups of $G$. Also, we investigate the characterizations for supersolvability of $\mathcal{F}_S (G)$ under the assumption that some subgroups of $G$ are strong $p$-$CAP$-subgroups of $G$.