%0 Journal Article
%T The Maschke property for the Sylow $p$-subgroups of the symmetric group $S_{p^n}$
%J International Journal of Group Theory
%I University of Isfahan
%Z 2251-7650
%A Green, David J.
%A Héthelyi, L.
%A Horváth, E.
%D 2018
%\ 12/01/2018
%V 7
%N 4
%P 41-64
%! The Maschke property for the Sylow $p$-subgroups of the symmetric group $S_{p^n}$
%K Maschke's Theorem
%K coprime action
%K Sylow $p$-subgroup of symmetric group
%K iterated wreath product
%K uniserial action
%R 10.22108/ijgt.2017.21610
%X 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.
%U https://ijgt.ui.ac.ir/article_21610_049d5dd2426c246d448583ee0a063476.pdf