The Maschke property for the Sylow $p$-subgroups of the symmetric group $S_{p^n}$
نویسندگان
1 Institut für Mathematik Friedrich-SchillerüUniversität 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.
doi
10.22108/ijgt.2017.21610چکیده
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.
