Wreath product is associative

From Groupprops
Jump to: navigation, search

Statement

Suppose G,H,K are groups. Suppose H comes equipped with an action on a set S and K comes equipped with an action on a set T. Note that, by the action of wreath product on Cartesian product, we can use this to define an action of the external wreath product H \wr K on S \times T. With these interpretations, we have that the external wreath product is associative up to isomorphism of groups:

(G \wr H) \wr K \cong G \wr (H \wr K)

Note that, to formulate this as an isomorphism of groups, we do not need to specify a group action of G on any set. However, if we do equip G with an action on a set A, then the two groups above have equivalent actions on the triple Cartesian product A \times S \times T using the action of wreath product on Cartesian product.