Some results on \(\pi\)-solvable and supersolvable groups (Q1321849)

The authors obtain conditions on certain maximal subgroups (or related concepts) of a finite group \(G\) that are sufficient (and sometimes necessary) for \(G\) to be \(p\)-solvable, \(\pi\)-solvable or supersolvable. For example (Theorem 3.1) if \(p\) is the largest prime factor of the order of \(G\) and \(G\) contains a maximal subgroup of composite \(p\)-free index, then \(G\) is \(p\)-solvable if and only if the normal index of \(M\) is \(p\)- free for every maximal subgroup \(M\) of \(G\) having composite \(p\)-free index.