Criteria for the p-solvability and p-supersolvability of finite groups

Let A, K, and H be subgroups of a group G and K ≤ H. Then we say that A covers the pair (K, H) if AH = AK and avoids the pair (K, H) if A ∩ H = A ∩ 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.