David Green - Institut für Mathematik Friedrich-SchillerüUniversität ۰۷۷۳۷ Jena
‎L. H&#۰۳۹;ethelyi - Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra H-۱۱۱۱ Budapest, Műegyetem rkp. ۳-۹.
E. Horv&#۰۳۹;ath - Budapest University of Technology and Economics, Faculty of Sciences, Inst. Math., Department of Algebra, H-۱۱۱۱ Budapest, Műegyetem rkp. ۳-۹.

‎‎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 $p>۳$ 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 $P_n$ of cyclic groups of order $p$‎, ‎i.e‎. ‎on Sylow $p$-subgroups of the symmetric groups $S_{p^n}$‎, ‎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 $P_n$ have a complement‎. ‎In the end we also describe abelian subgroups of $P_n$ of largest size‎.

‎Maschke's Theorem‎, ‎coprime action‎, ‎Sylow $p$-subgroup of symmetric group‎, ‎iterated wreath product‎, ‎uniserial action