Maximal subgroups of finite groups (Q5899826)

O. U. Kramer proved that a finite solvable group G is supersolvable (i.e., has rank 1) if, for every maximal subgroup M of G, the index of \(M\cap Fit G\) in the Fitting subgroup, Fit G, divides a prime. The author seeks to show that G has rank \(\leq 2\) if the index of \(M\cap Fit G\) in Fit G divides the square of a prime for every maximal subgroup M. He requires the additional hypothesis that G has odd order (Theorem 3.2) or is the direct product of a 2-group and a \(2'\)-group (Theorem 3.3). The work is based on a useful little lemma: The rank of finite solvable group G having trivial Frattini subgroup is bounded by the maximal value of i when, for every maximal subgroup M of G, [Fit G:M\(\cap Fit G]=p^ i\) for some prime p.