On Two Theorems of Finite Solvable Groups

Abstract

For a finite group G, let τ(G) denote a set of primes such that a prime p belongs to τ(G) if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions: (1) G has a p-complement for each p∈τ(G); (2) ∣τ(G)∣=2; (3) the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈τ(G); then G either is solvable or G/Sol(G)≅(2, 7) where Sol(G) is the largest solvable normal subgroup of (G).