In this paper we prove that every nonabelian finite 2-group with a cyclic commutator subgroup has a noninner automorphism of order two fixing either Φ(G) or Z(G) elementwise. This, together with a result of Peter Schmid on regular p-groups, extends our result to the class of nonabelian finite p-groups with a cyclic commutator subgroup.2010 Mathematics subject classification: primary 20D45; secondary 20D15.

Abstract. Let G be a non-trivial finite group and let A be a nilpotent subgroup of G. We prove that if jG : Aj c 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.