On nilpotent subgroups containing non-trivial normal subgroups

Abstract

Let G be a non-trivial finite group and let A be a nilpotent subgroup of G. We prove that if jG : Ajc expðAÞ, the exponent of A, then A contains a non-trivial normal subgroup of G. This extends an earlier result of Isaacs, who proved this in the case where A is abelian. We also show that if the above inequality is replaced by jG : Aj < ExpðGÞ, where ExpðGÞ denotes the order of a cyclic subgroup of G with maximal order, then A contains a non-trivial characteristic subgroup of G. We will use these results to derive some facts about transitive permutation groups.