Criteria for the $p$-Solvability and $p$-Supersolvability of Finite Groups

Let $A$, $K$, and $H$ be subgroups of a group $G$ and $K\leqslant H$. Then we say that $A$ covers the pair $(K,H)$ if $AH=AK$ and avoids the pair $(K,H)$ if $A\cap H=A\cap K$. A pair $(K,H)$ in $G$ is said to be maximal if $K$ is a maximal subgroup of $H$. In the present paper, we study finite groups in which some subgroups cover or avoid distinguished systems of maximal pairs of these groups. In particular, generalizations of a series of known results on (partial) $CAP$-subgroups are obtained.