# 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 A and H (sometimes known as the bottom and top[1]), there exist two variations of the wreath product: the unrestricted wreath product A Wr H and the restricted wreath product A wr H. The general form, denoted by A WrΩ H or A wrΩ H respectively, uses a set Ω with an H-action; when unspecified, usually Ω = H (a regular wreath product), though a different Ω is sometimes implied. The two variations coincide when A, H, and Ω are all finite. Either variation 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 is a central construction in the Krohn–Rhodes structure theory of finite semigroups.

## Definition

Let A and H be groups and Ω a set with H acting on it (from the left). Let K be the direct product

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

of copies of Aω := A indexed by the set Ω. The elements of K can be seen as arbitrary sequences (aω) 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

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

Then the unrestricted wreath product A WrΩ H of A by H is the semidirect product K ⋊ H. The subgroup K of A WrΩ H is called the base of the wreath product.

The restricted wreath product A wrΩ 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 K are sequences (aω) of elements in A indexed by Ω of which all but finitely many aω are the identity element of A.

In the most common case, one takes Ω := H, where H acts in a natural way on itself by left multiplication. In this case, the unrestricted and restricted wreath product may be denoted by A Wr H and A wr H respectively. This is called the regular wreath product.

## 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 to the circumstances.

• In literature AΩH may stand for the unrestricted wreath product A WrΩ H or the restricted wreath product A wrΩ H.
• Similarly, AH 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 = Sn is the symmetric group of degree n it is common in the literature to assume that Ω = {1,...,n} (with the natural action of Sn) and then omit Ω from the notation. That is, ASn commonly denotes A{1,...,n}Sn instead of the regular wreath product ASnSn. 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

### 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 A WrΩ H and the restricted wreath product A wrΩ H agree if the H-set Ω is finite. In particular this is true when Ω = H is finite.

### Subgroup

A wrΩ H is always a subgroup of A WrΩ H.

### Cardinality

If A, H and Ω are finite, then

|AΩH| = |A||Ω||H|.[2]

### Universal embedding theorem

Universal embedding theorem: If G is an extension of A by H, then there exists a subgroup of the unrestricted wreath product AH which is isomorphic to 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 A acts on a set Λ then there are two canonical ways to construct sets from Ω and Λ on which A WrΩ H (and therefore also A wrΩ H) can act.

• The imprimitive wreath product action on Λ × Ω.
If ((aω),h) ∈ A WrΩ H and (λ,ω′) ∈ Λ × Ω, then
${\displaystyle ((a_{\omega }),h)\cdot (\lambda ,\omega '):=(a_{h(\omega ')}\lambda ,h\omega ').}$
• The primitive wreath product action on ΛΩ.
An element in ΛΩ is a sequence (λω) indexed by the H-set Ω. Given an element ((aω), h) ∈ A WrΩ H its operation on (λω) ∈ ΛΩ is given by
${\displaystyle ((a_{\omega }),h)\cdot (\lambda _{\omega }):=(a_{h^{-1}\omega }\lambda _{h^{-1}\omega }).}$

## Examples

The base of this wreath product is the n-fold direct product
mn = ℤm × ... × ℤm
of copies of ℤm where the action φ : Sn → Aut(ℤmn) of the symmetric group Sn of degree n is given by
φ(σ)(α1,..., αn) := (ασ(1),..., ασ(n)).[5]
The action of Sn on {1,...,n} is as above. Since the symmetric group S2 of degree 2 is isomorphic to ℤ2 the hyperoctahedral group is a special case of a generalized symmetric group.[6]
• The smallest non-trivial wreath product is ℤ2≀ℤ2, which is the two-dimensional case of the above hyperoctahedral group. It is the symmetry group of the square, also called Dih4, the dihedral group of order 8.
• Let p be a prime and let n≥1. Let P be a Sylow p-subgroup of the symmetric group Spn. Then P is isomorphic to the iterated regular wreath product Wn = ℤp ≀ ℤp≀...≀ℤp of n copies of ℤp. Here W1 := ℤp and Wk := Wk−1≀ℤp for all k ≥ 2.[7][8] For instance, the Sylow 2-subgroup of S4 is the above ℤ2≀ℤ2 group.
• The Rubik's Cube group is a subgroup of index 12 in the product of wreath products, (ℤ3S8) × (ℤ2S12), 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 (S3S3) ≀ S2, where the factors are the permutation of rows/columns within a 3-row or 3-column band or stack (S3), the permutation of the bands/stacks themselves (S3) and the transposition, which interchanges the bands and stacks (S2). Here, the index sets Ω are the set of bands (resp. stacks) (|Ω| = 3) and the set {bands, stacks} (|Ω| = 2). Accordingly, |S3S3| = |S3|3|S3| = (3!)4 and |(S3S3) ≀ S2| = |S3S3|2|S2| = (3!)8 × 2.
• Wreath products arise naturally in the symmetry group of complete rooted trees and their graphs. For example, the repeated (iterated) wreath product S2S2...S2 is the automorphism group of a complete binary tree.

## References

1. ^ Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1998), "Wreath products", Notes on Infinite Permutation Groups, Lecture Notes in Mathematics, Berlin, Heidelberg: Springer, pp. 67–76, doi:10.1007/bfb0092558, ISBN 978-3-540-49813-1, retrieved 2021-05-12
2. ^ Joseph J. Rotman, An Introduction to the Theory of Groups, p. 172 (1995)
3. ^ 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)
4. ^ J D P Meldrum (1995). Wreath Products of Groups and Semigroups. Longman [UK] / Wiley [US]. p. ix. ISBN 978-0-582-02693-3.
5. ^ 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
6. ^ 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.
7. ^ Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995)
8. ^ 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)