Cover-avoidance properties and the structure of finite groups (Q1812027)

Let \(G\) be a finite group. The authors call a subgroup \(A\) of the group \(G\) a CAP-subgroup of \(G\) if for any chief factor \(H/K\) of \(G\) \(H\cap A=K\cap A\) or \(HA=KA\). Some characterizations for a finite group to be solvable are obtained when some of its maximal subgroups or \(2\)-maximal subgroups are CAP-subgroups. In particular, it is proved that a group \(G\) is solvable if and only if there exists a solvable \(2\)-maximal subgroup \(L\) of \(G\) such that \(L\) is a CAP-subgroup of \(G\). It is also proved that \(G\) is \(p\)-solvable if and only if a Sylow \(p\)-subgroup \(P\) of \(G\) is a CAP-subgroup of \(G\).