A note on \(p\)-solvable and solvable finite groups (Q1338443)

The authors show (Theorem 3.1) that a finite group is \(p\)-solvable, where \(p\) is the maximal prime factor of the order of \(G\), if and only if for every maximal subgroup \(M\) of composite index in \(G\), the \(p\)-factor of \([G:M]\) equals the \(p\)-factor of the normal index of \(M\) in \(G\). The authors also give a condition for solvability: Let \(\mathcal P\) be the set of maximal subgroups of \(G\) where indices are not divisible by \(p\) (defined as above); then \(G\) is solvable if each element of \(\mathcal P\) has prime index in \(G\).