# The Maschke property for the Sylow $p$-subgroups of the symmetric group $S_{p^n}$

Document Type : Research Paper

Authors

1 Institut f&uuml;r Mathematik Friedrich-Schiller&uuml;Universit&auml;t 07737 Jena

2 Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra H-1111 Budapest, Műegyetem rkp. 3-9.

3 Budapest University of Technology and Economics, Faculty of Sciences, Inst. Math., Department of Algebra, H-1111 Budapest, Műegyetem rkp. 3-9.

Abstract

‎‎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 $P_n$ 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.

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