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''), there exist two variations of the wreath product: the unrestricted wreath product $A \text H$ and the restricted wreath product $A \text H$. The general form, denoted by $A \text_ H$ or $A \text_ H$ respectively, requires that $H$ acts on some set $\Omega$; when unspecified, usually $\Omega = H$ (a regular wreath product), though a different $\Omega$ is sometimes implied. The two variations coincide when $A$, $H$, and $\Omega$ are all finite. Either variation is also denoted as $A \wr H$ (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 $A$ be a group and let $H$ be a group acting on a set $\Omega$ (on the left). The direct product $A^$ of $A$ with itself indexed by $\Omega$ is the set of sequences $\overline = (a_)_$ in $A$ indexed by $\Omega$, with a group operation given by pointwise multiplication. The action of $H$ on $\Omega$ can be extended to an action on $A^$ by ''reindexing'', namely by defining

: $h \cdot (a_)_ := (a_)_$

for all $h \in H$ and all $(a_)_ \in A^$.

Then the unrestricted wreath product $A \text_ H$ of $A$ by $H$ is the semidirect product $A^ \rtimes H$ with the action of $H$ on $A^$ given above. The subgroup $A^$ of $A^ \rtimes H$ is called the base of the wreath product.

The restricted wreath product $A \text_ H$ 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 $A$ with finitely-many non-identity entries.

In the most common case, $\Omega = H$, and $H$ acts on itself by left multiplication. In this case, the unrestricted and restricted wreath product may be denoted by $A \text H$ and $A \text H$ 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 ''A''≀_Ω''H'' may stand for the unrestricted wreath product ''A'' Wr_Ω ''H'' or the restricted wreath product ''A'' wr_Ω ''H''.
* Similarly, ''A''≀''H'' 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'' = ''S''_''n'' is the symmetric group of degree ''n'' it is common in the literature to assume that Ω =  (with the natural action of ''S''_''n'') and then omit Ω from the notation. That is, ''A''≀''S''_''n'' commonly denotes ''A''≀''S''_''n'' instead of the regular wreath product ''A''≀_''S''''n''''S''_''n''. 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'' Wr_Ω ''H'' and the restricted wreath product ''A'' wr_Ω ''H'' agree if the ''H''-set Ω is finite. In particular this is true when Ω = ''H'' is finite.

Subgroup

''A'' wr_Ω ''H'' is always a subgroup of ''A'' Wr_Ω ''H''.

Cardinality

If ''A'', ''H'' and Ω are finite, then
:: , ''A''≀_Ω''H'', = , ''A'', ^{, Ω,} , ''H'', .

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 ''A''≀''H'' which is isomorphic to ''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 ''A'' acts on a set Λ then there are two canonical ways to construct sets from Ω and Λ on which ''A'' Wr_Ω ''H'' (and therefore also ''A'' wr_Ω ''H'') can act.

* The imprimitive wreath product action on Λ × Ω.
*: If and , then
*:: $((a_\omega), h) \cdot (\lambda,\omega') := (a_\lambda, h\omega').$
* The primitive wreath product action on Λ^Ω.
*: An element in Λ^Ω is a sequence (''λ''_''ω'') indexed by the ''H''-set Ω. Given an element its operation on (''λ''_''ω'') ∈ Λ^Ω is given by
*:: $((a_\omega), h) \cdot (\lambda_\omega) := (a_\lambda_).$

Examples

* The Lamplighter group is the restricted wreath product ℤ₂≀ℤ.
* ( Generalized symmetric group).

: The base of this wreath product is the ''n''-fold direct product

:: ℤ_''m''^''n'' = ℤ_''m'' × ... × ℤ_''m''

: of copies of ℤ_''m'' where the action φ : ''S''_''n'' → Aut(ℤ_''m''^''n'') of the symmetric group ''S''_''n'' of degree ''n'' is given by

:: ''φ''(''σ'')(α₁,..., ''α''_''n'') := (''α''_''σ''(1),..., ''α''_{''σ''(''n'')}).

* ''S''₂≀''S''_''n'' (Hyperoctahedral group).

: The action of ''S''_''n'' on is as above. Since the symmetric group ''S''₂ of degree 2 is isomorphic to ℤ₂ the hyperoctahedral group is a special case of a generalized symmetric group.

* 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 ''Dih''₄, 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 ''S''_''p''^''n''. Then ''P'' is isomorphic to the iterated regular wreath product ''W''_''n'' = ℤ_''p'' ≀ ℤ_''p''≀...≀ℤ_''p'' of ''n'' copies of ℤ_''p''. Here ''W''₁ := ℤ_''p'' and ''W''_''k'' := ''W''_''k''−1≀ℤ_''p'' for all ''k'' ≥ 2.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) 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, (ℤ₃≀''S''₈) × (ℤ₂≀''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 (''S''₃ ≀ ''S''₃) ≀ ''S''₂, where the factors are the permutation of rows/columns within a 3-row or 3-column ''band'' or ''stack'' (''S''₃), the permutation of the bands/stacks themselves (''S''₃) and the transposition, which interchanges the bands and stacks (''S''₂). Here, the index sets ''Ω'' are the set of bands (resp. stacks) (, ''Ω'', = 3) and the set (, ''Ω'', = 2). Accordingly, , ''S''₃ ≀ ''S''₃, = , ''S''₃, ³, ''S''₃, = (3!)⁴ and , (''S''₃ ≀ ''S''₃) ≀ ''S''₂, = , ''S''₃ ≀ ''S''₃, ², ''S''₂, = (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 ''S''₂ ≀ ''S''₂ ≀ ''...'' ≀ ''S''₂ is the automorphism group of a complete binary tree.

References

External links

Wreath product
in '' Encyclopedia of Mathematics''.

Some Applications of the Wreath Product Construction
{{webarchive , url=https://web.archive.org/web/20140221081427/http://www.abstractmath.org/Papers/SAWPCWC.pdf , date=21 February 2014

Group products
Permutation groups
Binary operations