Wreath product is associative
Suppose are groups. Suppose comes equipped with an action on a set and comes equipped with an action on a set . Note that, by the action of wreath product on Cartesian product, we can use this to define an action of the external wreath product on . With these interpretations, we have that the external wreath product is associative up to isomorphism of groups:
Note that, to formulate this as an isomorphism of groups, we do not need to specify a group action of on any set. However, if we do equip with an action on a set , then the two groups above have equivalent actions on the triple Cartesian product using the action of wreath product on Cartesian product.