The lattice of periods of a group action
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.
scientific article; zbMATH DE number 745276