Wreath product

In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used in the classification of permutation groups and also provide a way of constructing interesting examples of groups.

Given two groups ${\displaystyle A}$ and ${\displaystyle H}$ (sometimes known as the bottom and top^[1]), there exist two variants of the wreath product: the unrestricted wreath product ${\displaystyle A{\text{ Wr }}H}$ and the restricted wreath product ${\displaystyle A{\text{ wr }}H}$. The general form, denoted by ${\displaystyle A{\text{ Wr}}_{\Omega }H}$ or ${\displaystyle A{\text{ wr}}_{\Omega }H}$ respectively, requires that ${\displaystyle H}$ acts on some set ${\displaystyle \Omega }$; when unspecified, usually ${\displaystyle \Omega =H}$ (a regular wreath product), though a different ${\displaystyle \Omega }$ is sometimes implied. The two variants coincide when ${\displaystyle A}$, ${\displaystyle H}$, and ${\displaystyle \Omega }$ are all finite. Either variant is also denoted as ${\displaystyle A\wr H}$ (with \wr for the LaTeX symbol) or A ≀ H (Unicode U+2240).

The notion generalizes to semigroups and, as such, is a central construction in the Krohn–Rhodes structure theory of finite semigroups.

Definition

Let ${\displaystyle A}$ be a group and let ${\displaystyle H}$ be a group acting on a set ${\displaystyle \Omega }$ (on the left). The direct product ${\displaystyle A^{\Omega }}$ of ${\displaystyle A}$ with itself indexed by ${\displaystyle \Omega }$ is the set of sequences ${\displaystyle {\overline {a}}=(a_{\omega })_{\omega \in \Omega }}$ in ${\displaystyle A}$, indexed by ${\displaystyle \Omega }$, with a group operation given by pointwise multiplication. The action of ${\displaystyle H}$ on ${\displaystyle \Omega }$ can be extended to an action on ${\displaystyle A^{\Omega }}$ by reindexing, namely by defining

${\displaystyle h\cdot (a_{\omega })_{\omega \in \Omega }:=(a_{h^{-1}\cdot \omega })_{\omega \in \Omega }}$

for all ${\displaystyle h\in H}$ and all ${\displaystyle (a_{\omega })_{\omega \in \Omega }\in A^{\Omega }}$.

Then the unrestricted wreath product ${\displaystyle A{\text{ Wr}}_{\Omega }H}$ of ${\displaystyle A}$ by ${\displaystyle H}$ is the semidirect product ${\displaystyle A^{\Omega }\rtimes H}$ with the action of ${\displaystyle H}$ on ${\displaystyle A^{\Omega }}$ given above. The subgroup ${\displaystyle A^{\Omega }}$ of ${\displaystyle A^{\Omega }\rtimes H}$ is called the base of the wreath product.

The restricted wreath product ${\displaystyle A{\text{ wr}}_{\Omega }H}$ 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 base consists of all sequences in ${\displaystyle A^{\Omega }}$ with finitely many non-identity entries. The two definitions coincide when ${\displaystyle \Omega }$ is finite.

In the most common case, ${\displaystyle \Omega =H}$, and ${\displaystyle H}$ acts on itself by left multiplication. In this case, the unrestricted and restricted wreath product may be denoted by ${\displaystyle A{\text{ Wr }}H}$ and ${\displaystyle A{\text{ wr }}H}$ respectively. This is called the regular wreath product.

Notation and conventions

The structure of the wreath product of ${\displaystyle A}$ by ${\displaystyle H}$ depends on the ${\displaystyle H}$-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 to the circumstances.

• In literature, ${\displaystyle A\wr _{\Omega }H}$ may stand for the unrestricted wreath product ${\displaystyle A\operatorname {Wr} _{\Omega }H}$ or the restricted wreath product ${\displaystyle A\operatorname {wr} _{\Omega }H}$.
• In literature, the ${\displaystyle H}$-set ${\displaystyle \Omega }$ may be omitted from the notation even if ${\displaystyle \Omega \neq H}$.
• In the special case that ${\displaystyle H=S_{n}}$ is the symmetric group of degree ${\displaystyle n}$, it is common in the literature to assume that ${\displaystyle \Omega =\{1,\dots ,n\}}$ (with the natural action of ${\displaystyle S_{n}}$) and then omit ${\displaystyle \Omega }$ from the notation. That is, ${\displaystyle A\wr S_{n}}$ commonly denotes ${\displaystyle A\wr _{\{1,\dots ,n\}}S_{n}}$ instead of the regular wreath product ${\displaystyle A\wr _{S_{n}}S_{n}}$. In the first case the base group is the product of ${\displaystyle n}$ copies of ${\displaystyle A}$, in the latter it is the product of n! copies of ${\displaystyle A}$.

Properties

Agreement of unrestricted and restricted wreath product on finite Ω

Since the finite direct product is the same as the finite direct sum of groups, it follows that the unrestricted wreath product ${\displaystyle A\operatorname {Wr} _{\Omega }H}$ and the restricted wreath product ${\displaystyle A\operatorname {wr} _{\Omega }H}$ are equal if ${\displaystyle \Omega }$ is finite. In particular, this is true when ${\displaystyle \Omega =H}$ and ${\displaystyle H}$ is finite.

Subgroup

${\displaystyle A\operatorname {wr} _{\Omega }H}$ is always a subgroup of ${\displaystyle A\operatorname {Wr} _{\Omega }H}$.

Cardinality

If ${\displaystyle A}$, ${\displaystyle H}$ and ${\displaystyle \Omega }$ are finite, then

${\displaystyle |A\wr _{\Omega }\!H|=|A|^{|\Omega |}|H|}$.^[2]

Universal embedding theorem

Main article: 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 ${\displaystyle G}$.^[3] This is also known as the Krasner–Kaloujnine embedding theorem. The Krohn–Rhodes theorem involves what is basically the semigroup equivalent of this.^[4]

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 ${\displaystyle A\operatorname {Wr} _{\Omega }H}$ (and therefore also ${\displaystyle A\operatorname {wr} _{\Omega }H}$) can act.

• The imprimitive wreath product action on ${\displaystyle \Lambda \times \Omega }$:

  If ${\displaystyle ((a_{\omega }),h)\in A\operatorname {Wr} _{\Omega }H}$ and ${\displaystyle (\lambda ,\omega ')\in \Lambda \times \Omega }$, then

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

• 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 ${\displaystyle ((a_{\omega }),h)\in A\operatorname {Wr} _{\Omega }H}$, its operation on ${\displaystyle (\lambda _{\omega })\in \Lambda ^{\Omega }}$ is given by

  ${\displaystyle ((a_{\omega }),h)\cdot (\lambda _{\omega }):=(a_{h^{-1}\omega }\lambda _{h^{-1}\omega }).}$

Examples

• The lamplighter group is the restricted wreath product ${\displaystyle C_{2}\wr \mathbb {Z} }$.

• The generalized symmetric group is ${\displaystyle C_{m}\wr S_{n}}$. The base of this wreath product is the ${\displaystyle n}$-fold direct product ${\displaystyle C_{m}^{n}=C_{m}\times \cdots \times C_{m}}$, where the action ${\displaystyle \phi :S_{n}\to {\text{Aut}}(C_{m}^{n})}$ of the symmetric group ${\displaystyle S_{n}}$ is given by ${\displaystyle \phi (\sigma )(\alpha _{1},\dots ,\alpha _{n})=(\alpha _{\sigma (1)},\dots ,\alpha _{\sigma (n)})}$.^[5]

• • As a special case, we have the hyperoctahedral group ${\displaystyle S_{2}\wr S_{n}}$ (since ${\displaystyle S_{2}}$ is isomorphic to ${\displaystyle C_{2}}$).^[6]

• The smallest non-trivial wreath product is ${\displaystyle C_{2}\wr C_{2}}$, which is the two-dimensional case of the above hyperoctahedral group. It is the symmetry group of the square, also called ${\displaystyle D_{8}}$, the dihedral group of order 8.

• Let ${\displaystyle p}$ be a prime and let ${\displaystyle n\geq 1}$. Let ${\displaystyle P}$ be a Sylow p-subgroup of the symmetric group ${\displaystyle S_{p^{n}}}$. Then ${\displaystyle P}$ is isomorphic to the iterated regular wreath product ${\displaystyle W_{n}=C_{p}\wr \cdots \wr C_{p}}$ of ${\displaystyle n}$ copies of ${\displaystyle C_{p}}$. Here ${\displaystyle W_{1}=C_{p}}$ and ${\displaystyle W_{k}=W_{k-1}\wr C_{p}}$ for all ${\displaystyle k\geq 2}$.^[7][8] For instance, the Sylow 2-subgroup of ${\displaystyle S_{4}}$ is the above ${\displaystyle C_{2}\wr C_{2}}$ group.

• The Rubik's Cube group is a normal subgroup of index 12 in the product of wreath products, ${\displaystyle (C_{3}\wr S_{8})\times (C_{2}\wr S_{12})}$, the factors corresponding to the symmetries of the 8 corners and 12 edges.

• The Sudoku validity-preserving transformations (VPT) group contains the double wreath product ${\displaystyle (S_{3}\wr S_{3})\wr S_{2}}$, where the factors are the permutation of rows/columns within a 3-row or 3-column band or stack (${\displaystyle S_{3}}$), the permutation of the bands/stacks themselves (${\displaystyle S_{3}}$) and the transposition, which interchanges the bands and stacks (${\displaystyle S_{2}}$). Here, the two index sets ${\displaystyle \Omega _{1},\Omega _{2}}$ are firstly the set of bands (resp. stacks), so ${\displaystyle |\Omega _{1}|=3}$, and secondly the set {bands, stacks} (so ${\displaystyle |\Omega _{2}|=2}$). Accordingly, ${\displaystyle |S_{3}\wr S_{3}|=|S_{3}|^{3}|S_{3}|=6^{4}}$ and ${\displaystyle |(S_{3}\wr S_{3})\wr S_{2}|=|S_{3}\wr S_{3}|^{2}|S_{2}|=6^{8}\times 2}$.

• Wreath products arise naturally in the symmetries of complete rooted trees and their graphs. For example, the repeated (iterated) wreath product ${\displaystyle S_{2}\wr S_{2}\wr \cdots \wr S_{2}}$ is the automorphism group of a complete binary tree.