# User:Mfluch/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 ${\displaystyle A}$ and ${\displaystyle H}$ there exists two variations of the wreath product: the unrestricted wreath product ${\displaystyle A{\mathop {\rm {Wr}}}H}$ (also written ${\displaystyle A\wr H}$) and the restricted wreath product ${\displaystyle A{\mathop {\rm {wr}}}H}$. Given a set ${\displaystyle \Omega }$ with an ${\displaystyle H}$-action there exists a generalisation of the wreath product which is denoted by ${\displaystyle A{\mathop {\rm {Wr}}}_{\Omega }H}$ or ${\displaystyle A{\mathop {\rm {wr}}}_{\Omega }H}$ respectively.

## Definition[1]

Let ${\displaystyle A}$ and ${\displaystyle H}$ be a groups and ${\displaystyle \Omega }$ a set with ${\displaystyle H}$ acting on it. Let ${\displaystyle K}$ be the direct product

${\displaystyle K:=\prod _{\omega \in \Omega }A_{\omega }}$

of copies of ${\displaystyle A_{\omega }:=A}$ indexed by the set ${\displaystyle \Omega }$. The elements of ${\displaystyle K}$ can be seen as arbitrary sequences ${\displaystyle (a_{\omega })}$ of elements of ${\displaystyle A}$ indexed by ${\displaystyle \Omega }$ with component wise multiplication. Then the action of ${\displaystyle H}$ on ${\displaystyle \Omega }$ extends in a natural way to an action of ${\displaystyle H}$ on the group ${\displaystyle K}$ by

${\displaystyle h\cdot (a_{\omega }):=(a_{h\omega })}$.

Then the unrestricted wreath product ${\displaystyle A{\mathop {\rm {Wr}}}_{\Omega }H}$ of ${\displaystyle A}$ by ${\displaystyle H}$ is the semidirect product ${\displaystyle K\rtimes H}$. The subgroup ${\displaystyle K}$ of ${\displaystyle A{\mathop {\rm {Wr}}}_{\Omega }H}$ is called the base of the wreath product.

The restricted wreath product ${\displaystyle A{\mathop {\rm {wr}}}_{\Omega }H}$ is constructed in the same way as the unrestricted wreath product except that one uses the direct sum

${\displaystyle K:=\bigoplus _{\omega \in \Omega }A_{\omega }}$

as the base of the wreath product. In this case the elements of ${\displaystyle K}$ are sequences ${\displaystyle (a_{\omega })}$ of elements in ${\displaystyle A}$ indexed by ${\displaystyle \Omega }$ of which all but finitely many ${\displaystyle a_{\omega }}$ are the identity element of ${\displaystyle A}$.

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

## Properties

• Since the finite direct product is the same as the finite direct sum of groups it follows that the unrestricted ${\displaystyle A{\mathop {\rm {Wr}}}_{\Omega }H}$ and the restricted wreath product ${\displaystyle A{\mathop {\rm {wr}}}_{\Omega }H}$ agree if ${\displaystyle \Omega }$ is finite. In particluar this is true when ${\displaystyle \Omega =H}$ is finite.
• Universal Embedding Theorem: If ${\displaystyle G}$ is an extension of ${\displaystyle A}$ by ${\displaystyle H}$, then there exists a subgroup of the unrestricted wreath product ${\displaystyle A\wr H}$ which is isomorphic to ${\displaystyle G}$.[2]
• If ${\displaystyle A}$, ${\displaystyle H}$ and ${\displaystyle \Omega }$ are finite, then
${\displaystyle |A\wr _{\Omega }H|=|A|^{|\Omega |}|H|}$.[3]
• From this follows immediately that in general the wreath product can neither be associative nor commutative.
For example, for finite groups ${\displaystyle T}$, ${\displaystyle D}$ and ${\displaystyle Q}$ it is easy to construct examples where
${\displaystyle |T\wr (D\wr Q)|\neq |(T\wr D)\wr Q|}$ and ${\displaystyle |T\wr Q|\neq |Q\wr T|}$.

## Canonical Actions of Wreath Products

If the group ${\displaystyle A}$ acts on a set ${\displaystyle \Lambda }$ then there are two canonical ways to construct sets from ${\displaystyle \Omega }$ and ${\displaystyle \Lambda }$ on which ${\displaystyle A{\mathop {\rm {Wr}}}_{\Omega }H}$ can act.

• The imprimitive wreath product action on ${\displaystyle \Lambda \times \Omega }$.
• The primitive wreath product action on ${\displaystyle \Lambda ^{\Omega }}$. An element in ${\displaystyle \Lambda ^{\Omega }}$ is a sequence ${\displaystyle (\lambda _{\omega })}$ indexed by the ${\displaystyle H}$-set ${\displaystyle \Omega }$. Given an element ${\displaystyle (a_{\omega },h)\in A{\mathop {\rm {Wr}}}H}$ its operation on ${\displaystyle (\lambda _{\omega })}$ is given by
${\displaystyle (a_{\omega },h)\cdot (\lambda _{\omega }):=(a_{h^{-1}\omega }\lambda _{h^{-1}\omega })}$.

## Examples

• The restricted wreath product ${\displaystyle \mathbb {Z} _{2}\,{\mathop {\rm {wr}}}\,\mathbb {Z} }$ is known as the Lamplighter group.
The base of this wreath product is the ${\displaystyle n}$-fold direct product
${\displaystyle (\mathbb {Z} _{m})^{n}=\mathbb {Z} _{m}\times \cdots \times \mathbb {Z} _{m}}$
of copies of ${\displaystyle \mathbb {Z} _{m}}$ where the action ${\displaystyle \varphi :S_{n}\to \operatorname {Aut} ((\mathbb {Z} _{m})^{n})}$ of symmetric group ${\displaystyle S_{n}}$ of degree n is given by
${\displaystyle \varphi (\sigma )(\alpha _{1},\ldots ,\alpha _{n}):=(\alpha _{\sigma (1)},\ldots ,\alpha _{\sigma (n)})}$.[4]
• ${\displaystyle S_{2}\wr _{\{1,\ldots ,n\}}S_{n}}$ (Hyperoctahedral group).
The action of ${\displaystyle S_{n}}$ on ${\displaystyle \{1,\ldots ,n\}}$ is as above. Since the symmetric group ${\displaystyle S_{2}}$ of degree 2 is isomorphic to ${\displaystyle \mathbb {Z} _{2}}$ the hyperoctahedral group is a special case of a geleralized symmetric group.[5]
• (Kaloujnine, 1948)[6] Let ${\displaystyle p}$ be a prime and let ${\displaystyle n\geq 1}$. Let ${\displaystyle P}$ be a Sylow ${\displaystyle p}$-subgroup of the symmetric group ${\displaystyle S_{p^{n}}}$ of degree ${\displaystyle p^{n}}$. Then ${\displaystyle P}$ is isomorphic to the iterated regular wreath product ${\displaystyle W_{n}=\mathbb {Z} _{p}\wr \mathbb {Z} _{p}\wr \cdots \wr \mathbb {Z} _{p}}$ of ${\displaystyle n}$ copies of ${\displaystyle \mathbb {Z} _{p}}$. Here ${\displaystyle W_{1}:=\mathbb {Z} }$ and ${\displaystyle W_{k}:=W_{k-1}\wr \mathbb {\mathbb {Z} } _{p}}$ for all ${\displaystyle k\geq 2}$.[7]

## References

1. ^ Joseph J. Rotman, An Introduction to the Theory of Groups, pp. 172ff. (1995)
2. ^ 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)
3. ^ Joseph J. Rotman, An Introduction to the Theory of Groups, p. 172 (1995)
4. ^ 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
5. ^ 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.
6. ^ 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)
7. ^ Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995)