# Wreath product

In mathematics, the **wreath product** of group theory is a specialized product of two groups, based on a semidirect product. Wreath products are an important tool in the classification of permutation groups and also provide a way of constructing interesting examples of groups.

Given two groups *A* and *H*, there exist two variations of the wreath product: the **unrestricted wreath product** *A* Wr *H* (also written *A*≀*H*) and the **restricted wreath product** *A* wr *H*. Given a set Ω with an *H*-action there exists a generalisation of the wreath product which is denoted by *A* Wr_{Ω} *H* or *A* wr_{Ω} *H* respectively.

## Definition

Let *A* and *H* be groups and Ω a set with *H* acting on it. Let *K* be the direct product

of copies of *A*_{ω} := *A* indexed by the set Ω. The elements of *K* can be seen as arbitrary sequences (*a*_{ω}) of elements of *A* indexed by Ω with component wise multiplication. Then the action of *H* on Ω extends in a natural way to an action of *H* on the group *K* by

Then the **unrestricted wreath product** *A* Wr_{Ω} *H* of *A* by *H* is the semidirect product *K* ⋊ *H*. The subgroup *K* of *A* Wr_{Ω} *H* is called the **base** of the wreath product.

The **restricted wreath product** *A* wr_{Ω} *H* is constructed in the same way as the unrestricted wreath product except that one uses the direct sum

as the base of the wreath product. In this case the elements of *K* are sequences (*a*_{ω}) of elements in *A* indexed by Ω of which all but finitely many *a*_{ω} are the identity element of *A*.

The group *H* acts in a natural way on itself by left multiplication. Thus we can choose Ω := *H*. In this special (but very common) case the unrestricted and restricted wreath product may be denoted by *A* Wr *H* and *A* wr *H* respectively. We say in this case that the wreath product is **regular**.

## Notation and Conventions

The structure of the wreath product of *A* by *H* depends on the *H*-set Ω and in case Ω is infinite it also depends on whether one uses the restricted or unrestricted wreath product. However, in literature the notation used may be deficient and one needs to pay attention on the circumstances.

- In literature
*A*≀_{Ω}*H*may stand for the unrestricted wreath product*A*Wr_{Ω}*H*or the restricted wreath product*A*wr_{Ω}*H*.

- Similarly,
*A*≀*H*may stand for the unrestricted regular wreath product*A*Wr*H*or the restricted regular wreath product*A*wr*H*.

- In literature the
*H*-set Ω may be omitted from the notation even if Ω≠H.

- In the special case that
*H*=*S*_{n}is the symmetric group of degree*n*it is common in the literature to assume that Ω={1,...,*n*} (with the natural action of*S*_{n}) and then omit Ω from the notation. That is,*A*≀*S*_{n}commonly denotes*A*≀_{{1,...,n}}*S*_{n}instead of the regular wreath product*A*≀_{Sn}*S*_{n}. In the first case the base group is the product of*n*copies of*A*, in the latter it is the product of*n*! copies of*A*.

## Properties

- Since the finite direct product is the same as the finite direct sum of groups, it follows that the unrestricted
*A*Wr_{Ω}*H*and the restricted wreath product*A*wr_{Ω}*H*agree if the*H*-set Ω is finite. In particular this is true when Ω =*H*is finite.

*A*wr_{Ω}*H*is always a subgroup of*A*Wr_{Ω}*H*.

- Universal Embedding Theorem: If
*G*is an extension of*A*by*H*, then there exists a subgroup of the unrestricted wreath product*A*≀*H*which is isomorphic to*G*.^{[1]}

- If
*A*,*H*and Ω are finite, then

- |
*A*≀_{Ω}*H*| = |*A*|^{|Ω|}|*H*|.^{[2]}

- |

## Canonical Actions of Wreath Products

If the group *A* acts on a set Λ then there are two canonical ways to construct sets from Ω and Λ on which *A* Wr_{Ω} *H* (and therefore also *A* wr_{Ω} *H*) can act.

- The
**imprimitive**wreath product action on Λ×Ω.

- If ((
*a*_{ω}),*h*)∈*A*Wr_{Ω}*H*and (λ,ω')∈Λ×Ω, then

- The
**primitive**wreath product action on Λ^{Ω}.

- An element in Λ
^{Ω}is a sequence (λ_{ω}) indexed by the*H*-set Ω. Given an element ((*a*_{ω}),*h*) ∈*A*Wr_{Ω}*H*its operation on (λ_{ω})∈Λ^{Ω}is given by

## Examples

- The Lamplighter group is the restricted wreath product ℤ
_{2}≀ℤ.

- ℤ
_{m}≀*S*_{n}(Generalized symmetric group).

- The base of this wreath product is the
*n*-fold direct product

- ℤ
_{m}^{n}= ℤ_{m}× ... × ℤ_{m}

- ℤ

- of copies of ℤ
_{m}where the action φ :*S*_{n}→ Aut(ℤ_{m}^{n}) of the symmetric group*S*_{n}of degree*n*is given by

- φ(σ)(α
_{1},..., α_{n}) := (α_{σ(1)},..., α_{σ(n)}).^{[3]}

- φ(σ)(α

*S*_{2}≀*S*_{n}(Hyperoctahedral group).

- The action of
*S*_{n}on {1,...,*n*} is as above. Since the symmetric group*S*_{2}of degree 2 is isomorphic to ℤ_{2}the hyperoctahedral group is a special case of a generalized symmetric group.^{[4]}

- Let
*p*be a prime and let*n*≥1. Let*P*be a Sylow*p*-subgroup of the symmetric group*S*_{pn}of degree*p*^{n}. Then*P*is isomorphic to the iterated regular wreath product*W*_{n}= ℤ_{p}≀ ℤ_{p}≀...≀ℤ_{p}of*n*copies of ℤ_{p}. Here*W*_{1}:= ℤ_{p}and*W*_{k}:=*W*_{k-1}≀ℤ_{p}for all*k*≥2.^{[5]}^{[6]}

- The Rubik's Cube group is a subgroup of small index in the product of wreath products, (ℤ
_{3}≀*S*_{8}) × (ℤ_{2}≀*S*_{12}), the factors corresponding to the symmetries of the 8 corners and 12 edges.

## References

- ↑ M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de groupes III", Acta Sci. Math. Szeged 14, pp. 69-82 (1951)
- ↑ Joseph J. Rotman, An Introduction to the Theory of Groups, p. 172 (1995)
- ↑ J. W. Davies and A. O. Morris, "The Schur Multiplier of the Generalized Symmetric Group", J. London Math. Soc (2), 8, (1974), pp. 615-620
- ↑ P. Graczyk, G. Letac and H. Massam, "The Hyperoctahedral Group, Symmetric Group Representations and the Moments of the Real Wishart Distribution", J. Theoret. Probab. 18 (2005), no. 1, 1-42.
- ↑ Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995)
- ↑ L. Kaloujnine, "La structure des p-groupes de Sylow des groupes symétriques finis", Annales Scientifiques de l'École Normale Supérieure. Troisième Série 65, pp. 239–276 (1948)