Finite solvable groups with a nilpotent normal complement subgroup

Abstract

Let [Formula: see text] be a finite solvable group and [Formula: see text] a non-normal core-free solvable subgroup of [Formula: see text]. We show that if the normalizer of any nontrivial normal subgroup of [Formula: see text] is equal to [Formula: see text], then [Formula: see text] has a nilpotent normal complement [Formula: see text] such that [Formula: see text] and [Formula: see text] is a Frobenius group.