The lattice of periods of a group action
type: Item
instance of: scholarly article
title: The lattice of periods of a group action
zbMATH Open document ID: 0823.05006
author: William F. IV Doran
published in: Advances in Mathematics
publication date: 1995-10-23T00:00:00Z
review text: Let \(G\) be a finite group and \(S\) a finite set. An action of \(G\) on \(S\) yields a partition of \(S\) given by the orbits of the action. By restricting the action to subgroups, one obtains an order-preserving map from the lattice of subgroups of \(G\) to the lattice of partitions of \(S\). The image of this map is called the lattice of periods of the group action; it is indeed a lattice. The author presents a fairly comprehensive study of this construction. The main result of the paper is that the lattice of periods depends only on the irreducible \(G\)-modules appearing in the permutation representation determined by the action, and not on their multiplicities. The author also treats the problem of determining which posets arise as lattices of periods for \(G\)-actions. A method is given for producing all such posets and is carried out in case \(G\) is an abelian group. The paper closes with an extended example.
Mathematics Subject Classification ID: 05A18
Mathematics Subject Classification ID: 05E99
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 745276
zbMATH Keywords: lattice of subgroups
zbMATH Keywords: lattice of partitions
zbMATH Keywords: lattice of periods of the group action
zbMATH Keywords: permutation representation
zbMATH Keywords: posets
zbMATH Keywords: abelian group
reviewed by: Michael Falk
MaRDI profile type: Publication
full work available at URL: aima.1995.1003
OpenAlex ID: W2023513867
DOI: 10.1006/AIMA.1995.1003
description: scientific article; zbMATH DE number 745276