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

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


 * $$K := \prod_{\omega \in \Omega} A_\omega$$

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


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

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

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


 * $$K := \bigoplus_{\omega \in \Omega} A_\omega$$

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

The group $$H$$ acts in a natural way on itself by left multiplication. Thus we can choose $$\Omega := H$$. In this special (but very common) case the unrestricted and restricted wreath product may be denoted by $$A \mathop{\rm Wr} H$$ and $$A \mathop{\rm wr} H$$ 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 $$A\mathop{\rm Wr}_\Omega H$$ and the restricted wreath product $$A\mathop{\rm wr}_\Omega H$$ agree if $$\Omega$$ is finite. In particluar this is true when $$\Omega = H$$ is finite.


 * Universal Embedding Theorem: If $$G$$ is an extension of $$A$$ by $$H$$, then there exists a subgroup of the unrestricted wreath product $$A \wr H$$ which is isomorphic to $$G$$.


 * If $$A$$, $$H$$ and $$\Omega$$ are finite, then


 * $$|A\wr_\Omega H| = |A|^{|\Omega|}|H|$$.


 * From this follows immediately that in general the wreath product can neither be associative nor  commutative.


 * For example, for finite groups $$T$$, $$D$$ and $$Q$$ it is easy to construct examples where


 * $$|T\wr (D\wr Q)| \neq |(T\wr D)\wr Q|$$ and $$|T\wr Q| \neq |Q\wr T|$$.

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


 * The imprimitive wreath product action on $$\Lambda\times \Omega$$.


 * The primitive wreath product action on $$\Lambda^\Omega$$. An element in $$\Lambda^\Omega$$ is a sequence $$(\lambda_\omega)$$ indexed by the $$H$$-set $$\Omega$$. Given an element $$(a_\omega, h)\in A\mathop{\rm Wr} H$$ its operation on $$(\lambda_\omega)$$ is given by


 * $$(a_\omega, h) \cdot (\lambda_\omega) := (a_{h^{-1}\omega}\lambda_{h^{-1}\omega})$$.

Examples

 * The restricted wreath product $$\mathbb{Z}_2\, \mathop{\rm wr}\, \mathbb{Z}$$ is known as the Lamplighter group.


 * $$\mathbb{Z}_m \wr_{\{1,\ldots,n\}} S_n$$ (Generalized symmetric group).


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


 * $$(\mathbb{Z}_m)^n = \mathbb{Z}_m \times \cdots \times \mathbb{Z}_m$$


 * of copies of $$\mathbb{Z}_m$$ where the action $$\varphi : S_n \to \operatorname{Aut}((\mathbb{Z}_m)^n)$$ of symmetric group $$S_n$$ of degree n is given by


 * $$\varphi(\sigma)(\alpha_1, \ldots, \alpha_n) := (\alpha_{\sigma(1)}, \ldots, \alpha_{\sigma(n)})$$.


 * $$S_2 \wr_{\{1,\ldots,n\}} S_n$$ (Hyperoctahedral group).


 * The action of $$S_n$$ on $$\{1,\ldots, n\}$$ is as above. Since the symmetric group $$S_2$$ of degree 2 is isomorphic to $$\mathbb{Z}_2$$ the hyperoctahedral group is a special case of a geleralized symmetric group.


 * (Kaloujnine, 1948) Let $$p$$ be a prime and let $$n\geq 1$$. Let $$P$$ be a  Sylow $p$-subgroup of the symmetric group $$S_{p^n}$$ of degree $$p^n$$. Then $$P$$ is  isomorphic to the iterated regular wreath product $$W_n = \mathbb{Z}_p \wr \mathbb{Z}_p \wr \cdots \wr \mathbb{Z}_p$$ of $$n$$ copies of $$\mathbb{Z}_p$$. Here $$W_1 := \mathbb{Z}$$ and $$W_k := W_{k-1}\wr \mathbb{\Z}_p$$ for all $$k\geq 2$$.