On CAS-subgroups of finite groups. (Q995390)

Let \(G\) be a finite group and \(H\) a normal subgroup of \(G\). A subgroup of \(G\) is defined to be a CAS-subgroup of \(G\) if it has a complement that has the covering-avoidance property. Theorem 2.5: \(G\) is soluble if and only if the Sylow 2- and 3-subgroups of \(G\) are CAS-subgroups of \(G\). -- Theorem 3.1: If \(G/H\) is nilpotent, \(p\) a prime dividing the order of \(G\), while the order of \(G\) and \(p-1\) are relatively prime, \(P\) a Sylow \(p\)-subgroup of \(H\) and all maximal subgroups of \(P\) are CAS-subgroups, then \(G\) is \(p\)-nilpotent. -- Theorem 4.3: If \(G/H\) is supersoluble and the Sylow subgroups of \(H\) are CAS-subgroups of \(G\), then \(G\) is supersoluble.