In _{Ω} ''H'' or ''A'' wr_{Ω} ''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 $A\; \backslash wr\; H$ (with \wr for the LaTeX symbol) or ''A'' ≀ ''H'' (

_{ω} := ''A'' indexed by the set Ω. The elements of ''K'' can be seen as arbitrary _{ω}) 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
: $h\; (a\_\backslash omega)\; :=\; (a\_).$
Then the unrestricted wreath product ''A'' Wr_{Ω} ''H'' of ''A'' by ''H'' is the _{Ω} ''H'' is called the base of the wreath product.
The restricted wreath product ''A'' wr_{Ω} ''H'' is constructed in the same way as the unrestricted wreath product except that one uses the _{ω}) of elements in ''A'' indexed by Ω of which all but finitely many ''a''_{ω} are the

_{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.
* (_{''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''} (_{''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 _{''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 _{2} ≀ ''S''_{2} ≀ ''...'' ≀ ''S''_{2} is the automorphism group of a complete

Wreath product

in ''

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 theory Permutation groups Binary operations

group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, the wreath product is a special combination of two groups
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic identi ...

based on the semidirect product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

. It is formed by the action
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of one group on many copies of another group, somewhat analogous to exponentiation
Exponentiation is a mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

. Wreath products are used in the classification of permutation group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s 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'' Wr ''H'' and the restricted wreath product ''A'' wr ''H''. The general form, denoted by ''A'' WrUnicode
Unicode, formally the Unicode Standard, is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's wri ...

U+2240).
The notion generalizes to semigroup
In mathematics, a semigroup is an algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

s 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
Acting is an activity in which a story is told by means of its Enactment (psychology), enactment by an actor or actress who adopts a Character (arts), character—in theatre, television, film, radio, or any other medium that makes use of the ...

on it (from the left). Let ''K'' be the direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

: $K\; =\; \backslash prod\_\; A\_\backslash omega$
of copies of ''A''sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s (''a''semidirect product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

''K'' ⋊ ''H''. The subgroup ''K'' of ''A'' Wrdirect sum
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

: $K\; =\; \backslash bigoplus\_\; A\_\backslash omega$
as the base of the wreath product. In this case the elements of ''K'' are sequences (''a''identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

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 ''A''≀symmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

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, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

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 ℤGeneralized symmetric group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

).
: The base of this wreath product is the ''n''-fold direct product
:: ℤsymmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

''S''Hyperoctahedral group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

).
: The action of ''S''isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to ℤdihedral group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

of order 8.
* Let ''p'' be a prime
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

and let ''n''≥1. Let ''P'' be a Sylow ''p''-subgroup of the symmetric group ''S''isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to the iterated regular wreath product ''W''trees
In botany
Botany, also called , plant biology or phytology, is the science
Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organiz ...

and their graphs
Graph may refer to:
Mathematics
*Graph (discrete mathematics)
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes an ...

. For example, the repeated (iterated) wreath product ''S''binary tree
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , ...

.
References

External links

Wreath product

in ''

Encyclopedia of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics.
Overview
The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate ...

''.
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 theory Permutation groups Binary operations