The Maschke property for the Sylow \(p\)-subgroups of the symmetric group \(S_{p^n}\). (English) Zbl 1446.20035
Summary: 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.
MSC:
| 20D45 | Automorphisms of abstract finite groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20B35 | Subgroups of symmetric groups |
Keywords:
Maschke’s theorem; coprime action; Sylow \(p\)-subgroup of symmetric group; iterated wreath product; uniserial actionReferences:
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