Action of wreath product on Cartesian product

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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 Cartesian product A \times S as follows.

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 \times S as follows: An element of G^S is a function f:S \to G. For an element (a,s), the action of f on (a,s) is obtained as (f(s) \cdot a, s) where f(s) \cdot a is the action of the element f(s) \in G on A. In other words, it permutes each of the fibers according to the f-value on that fiber.
  • H acts on A \times S as follows: An element h \in H sends (a,s) \in A \times S to (a,h \cdot s) where h \cdot s denotes the image of s under the action of h.

Note that we need to verify a compatibility condition to show that this is well defined.