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This article describes a product notion for groups

Contents 1 Definition 1.1 For an abstract group and a group of permutations 1.2 For two abstract groups 1.3 Equivalence with the internal wreath product 2 Facts 2.1 The wreath product of two groups of permutations 2.2 Associativity of wreath product

Definition

For an abstract group and a group of permutations

Let G be any abstract group and H be a group along with a homomorphism for some set S (in other words, we are given a permutation representation, or group action, of H). Then, the external wreath product of G by H (relative to ρ) is defined as the external semidirect product of the group G^S by H, where an element sends to the function .

This wreath product is typically denoted as:

The group G is termed the base of the wreath product.

For two abstract groups

Let G,H be abstract groups. Then, the external wreath product of G by H is typically understood as the wreath product where the action of H is taken to be the left-regular action on itself as a set. In other words, the homomorphism ρ is the natural embedding arising via Cayley's theorem.

Equivalence with the internal wreath product

An internal wreath product is the equivalent to an external wreath product, except that here we start with the whole group, and find subgroups G and H such that the given group is an internal semidirect product of a direct product of copies of G by H, acting via coordinate permutations only. A subgroup G arising in this way is termed a base of a wreath product.

Facts

The wreath product of two groups of permutations

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, acts naturally on two sets: the Cartesian product , and the function space A^S of functions from S to A.

Further information: Action of wreath product on Cartesian product, Action of wreath product on function space

Associativity of wreath product

The associativity law:

holds with a suitable interpretation. For instance, if we are given the left side, we need to justify how is to be viewed as a group of permutations for the right side to make sense. Both H and K come as groups of permutations, and we equip with the induced action on the Cartesian product of the spaces they're acting on.

Further information: Wreath product is associative