p-Supersolvability and actions on p-groups stabilizing certain subgroups

Let G be a finite group having a noncyclic Sylow p-subgroup of order exceeding p(e), where e >= 3. If every noncyclic subgroup of G of order p(e) is normal in G, we show that G is p-supersolvable, and in fact we prove this under the much weaker hypothesis that the noncyclic subgroups of order p(e) are S-semipermutable in G. The key to the proof is to study the action of a group A on a p-group P under the condition that every noncyclic subgroup of P with order p(e) is stabilized by A. (C) 2014 Elsevier Inc. All rights reserved.