The Maschke property for the Sylow p-groups of the symmetric group Spn

‎‎In this paper we prove that the Maschke property holds for coprime actions on some important classes of p-groups like‎: ‎metacyclic p-groups‎, ‎p-groups of p-rank two for p 3 and some weaker property holds in the case of regular p-groups‎. ‎The main focus will be the case of coprime actions on the iterated wreath product Pn of cyclic groups of order p‎, ‎i.e‎. ‎on Sylow p-subgroups of the symmetric groups Spn‎, ‎where we also prove that a stronger form of the Maschke property holds‎. ‎These results contribute to a future possible classification of all p-groups with the Maschke property‎. ‎We apply these results to describe which normal partition subgroups of Pn have a complement‎. ‎In the end we also describe abelian subgroups of Pn of largest size‎.