Wreath product

Notation and conventions

The structure of the wreath product of ${\displaystyle A}$ by ${\displaystyle H}$ depends on the ${\displaystyle H}$-set ${\displaystyle \mathrm{\Omega }}$ and in case ${\displaystyle \mathrm{\Omega }}$ 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, ${\displaystyle A{\wr }_{\mathrm{\Omega }}H}$ may stand for the unrestricted wreath product ${\displaystyle A{\mathrm{Wr}}_{\mathrm{\Omega }}H}$ or the restricted wreath product ${\displaystyle A{\mathrm{wr}}_{\mathrm{\Omega }}H}$.
• In literature, the ${\displaystyle H}$-set ${\displaystyle \mathrm{\Omega }}$ may be omitted from the notation even if ${\displaystyle \mathrm{\Omega }\ne H}$.
• 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 ${\displaystyle \mathrm{\Omega }=\{1,\dots ,n\}}$ (with the natural action of ${\displaystyle S_n}$) and then omit ${\displaystyle \mathrm{\Omega }}$ 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 ${\displaystyle A{\mathrm{Wr}}_{\mathrm{\Omega }}H}$ and the restricted wreath product ${\displaystyle A{\mathrm{wr}}_{\mathrm{\Omega }}H}$ are equal if ${\displaystyle \mathrm{\Omega }}$ is finite. In particular, this is true when ${\displaystyle \mathrm{\Omega }=H}$ and ${\displaystyle H}$ is finite.

Subgroup

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

Cardinality

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

${\displaystyle |A{\wr }_{\mathrm{\Omega }}H|=|A{|}^{|\mathrm{\Omega }|}|H|}$.

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 ${\displaystyle G}$. This is also known as the Krasner–Kaloujnine embedding theorem. The Krohn–Rhodes theorem involves what is basically the semigroup equivalent of this.

Canonical actions of wreath products

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

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

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

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

• The primitive wreath product action on ${\displaystyle {\mathrm{\Lambda }}^{\mathrm{\Omega }}}$:

  An element in ${\displaystyle {\mathrm{\Lambda }}^{\mathrm{\Omega }}}$ is a sequence ${\displaystyle ({\lambda }_{\omega })}$ indexed by the ${\displaystyle H}$-set ${\displaystyle \mathrm{\Omega }}$. Given an element ${\displaystyle ((a_{\omega }),h)\in A{\mathrm{Wr}}_{\mathrm{\Omega }}H}$, its operation on ${\displaystyle ({\lambda }_{\omega })\in {\mathrm{\Lambda }}^{\mathrm{\Omega }}}$ 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 ${\displaystyle \varphi :S_n\to \text{Aut}(C_m^n)}$ of the symmetric group ${\displaystyle S_n}$ is given by ${\displaystyle \varphi (\sigma )({\alpha }_1,\dots ,{\alpha }_n)=({\alpha }_{\sigma (1)},\dots ,{\alpha }_{\sigma (n)})}$.
  • As a special case, we have the hyperoctahedral group ${\displaystyle S_2\wr S_n}$ (since ${\displaystyle S_2}$ is isomorphic to ${\displaystyle C_2}$).
• 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 ${\displaystyle n\ge 1}$. 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 ${\displaystyle k\ge 2}$. 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 ${\displaystyle {\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2}$ are firstly the set of bands (resp. stacks), so ${\displaystyle |{\mathrm{\Omega }}_1|=3}$, and secondly the set {bands, stacks} (so ${\displaystyle |{\mathrm{\Omega }}_2|=2}$). 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.