On generalized covering and avoidance properties of finite groups and saturated fusion systems

A subgroup $A$ of a finite group $G$ is said to be a $CAP$-subgroup of $G$, if for any chief factor $H\mathrm{/}K$ of $G$, either $AH=AK$ or $A\cap H=A\cap K$. 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 $1=S_0\le S_1\le \cdots \le S_n=S$, such that $S_{i+1}\mathrm{/}{S}_i$ is cyclic, and $S_i$ is strongly $\mathcal{F}$-closed for any $i=0,1,\cdots ,n$. In this paper, we first introduce the concept of strong $p$-$CAP$-subgroups, and investigate the structure of finite groups under the assumptions that some subgroups of $G$ are partial $CAP$-subgroups or strong $(p)$-$CAP$-subgroups of $G$, and obtain some criteria for a group $G$ to be $p$-supersolvable. After that, we investigate the characterizations for supersolvability of ${\mathcal{F}}_S(G)$ under the assumptions that some subgroups of $G$ are partial $CAP$-subgroups or strong $(p)$-$CAP$-subgroups of $G$, and obtain some criteria for a fusion system ${\mathcal{F}}_S(G)$ to be supersolvable. The above results improve some known results and develop some new results about $CAP$-subgroups from fusion systems.