Action of wreath product on function space

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Definition

Suppose both G and H are being viewed along with specific group actions of each, i.e., they are both being viewed as groups of permutations, with G acting on a set A and H acting on a set S. Then, G \wr H comes equipped with a natural action on the set A^S of all functions from S to A.

First, recall that G \wr S is the external semidirect product G^S \rtimes H. To specify an action of this, we will specify how G^S and H act; then, via the equivalence of internal and external semidirect product, we would have defined the action of a generating set and hence of all of G \wr H. Here is the action:

  • G^S acts on A^S as follows: Given a function f: S \to G and a function \theta:S \to A, define f \cdot \theta as the function S \to A given by s \mapsto f(s) \cdot \theta(s), where we use the action of G on A. One way of thinking of this is that the action is coordinate-wise or point-wise.
  • H acts on A^S as follows: Given an element h \in H and a function \theta:S \to A, we define h \cdot \theta as the map s \mapsto \theta(h^{-1} \cdot s) where we use the action of H on S on the inside.