In

_{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 _{''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''.

_{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.

_{Ω} ''H'' is always a _{Ω} ''H''.

_{Ω}''H'', = , ''A'', ^{, Ω, }, ''H'', .

_{Ω} ''H'' (and therefore also ''A'' wr_{Ω} ''H'') can act.
* The imprimitive wreath product action on Λ × Ω.
*: If and , then
*:: $((a\_\backslash omega),\; h)\; \backslash cdot\; (\backslash lambda,\backslash omega\text{'})\; :=\; (a\_\backslash lambda,\; h\backslash omega\text{'}).$
* 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
*:: $((a\_\backslash omega),\; h)\; \backslash cdot\; (\backslash lambda\_\backslash omega)\; :=\; (a\_\backslash lambda\_).$

_{2}≀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 _{''n''} of degree ''n'' is given by
:: ''φ''(''σ'')(α_{1},..., ''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 _{2} the hyperoctahedral group is a special case of a generalized symmetric group.
* The smallest non-trivial wreath product is ℤ_{2}≀ℤ_{2}, 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 _{''p''''n''}. Then ''P'' is _{''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 S_{4} is the above ℤ_{2}≀ℤ_{2} group.
* The Rubik's Cube group is a subgroup of index 12 in the product of wreath products, (ℤ_{3}≀''S''_{8}) × (ℤ_{2}≀''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

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

group theory
In abstract algebra
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. The ...

, 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
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...

. 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 $A\; \backslash text\; H$ and the restricted wreath product $A\; \backslash text\; H$. The general form, denoted by $A\; \backslash text\_\; H$ or $A\; \backslash text\_\; H$ respectively, requires that $H$ acts on some set $\backslash Omega$; when unspecified, usually $\backslash Omega\; =\; H$ (a regular wreath product), though a different $\backslash Omega$ is sometimes implied. The two variations coincide when $A$, $H$, and $\backslash Omega$ are all finite. Either variation is also denoted as $A\; \backslash wr\; H$ (with \wr for the LaTeX symbol) or ''A'' ≀ ''H'' (Unicode
Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...

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 $\backslash Omega$ (on the left). The direct product $A^$ of $A$ with itself indexed by $\backslash Omega$ is the set of sequences $\backslash overline\; =\; (a\_)\_$ in $A$ indexed by $\backslash Omega$, with a group operation given by pointwise multiplication. The action of $H$ on $\backslash Omega$ can be extended to an action on $A^$ by ''reindexing'', namely by defining : $h\; \backslash cdot\; (a\_)\_\; :=\; (a\_)\_$ for all $h\; \backslash in\; H$ and all $(a\_)\_\; \backslash in\; A^$. Then the unrestricted wreath product $A\; \backslash text\_\; H$ of $A$ by $H$ is the semidirect product $A^\; \backslash rtimes\; H$ with the action of $H$ on $A^$ given above. The subgroup $A^$ of $A^\; \backslash rtimes\; H$ is called the base of the wreath product. The restricted wreath product $A\; \backslash text\_\; H$ is constructed in the same way as the unrestricted wreath product except that one uses thedirect sum
The direct sum is an operation between structures in abstract algebra
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are ...

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, $\backslash Omega\; =\; H$, and $H$ acts on itself by left multiplication. In this case, the unrestricted and restricted wreath product may be denoted by $A\; \backslash text\; H$ and $A\; \backslash text\; H$ 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''≀symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a ...

of degree ''n'' it is common in the literature to assume that Ω = (with the natural action of ''S''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'' WrSubgroup

''A'' wrsubgroup
In group theory, a branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These ...

of ''A'' WrCardinality

If ''A'', ''H'' and Ω are finite, then :: , ''A''≀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'' WrExamples

* The Lamplighter group is the restricted wreath product ℤsymmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a ...

''S''isomorphic
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

to ℤprime
A prime number (or a prime) is a natural number
In mathematics, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3 ...

and let ''n''≥1. Let ''P'' be a Sylow ''p''-subgroup of the symmetric group ''S''isomorphic
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

to the iterated regular wreath product ''W''binary tree
In computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to prac ...

.
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