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In
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
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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'' 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 A \wr H (with \wr for the LaTeX symbol) or ''A'' ≀ ''H'' (
Unicode 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 ...

Unicode
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 = \prod_ A_\omega of copies of ''A''ω := ''A'' indexed by the set Ω. The elements of ''K'' can be seen as arbitrary
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 ...

sequence
s (''a''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_\omega) := (a_). Then the unrestricted wreath product ''A'' WrΩ ''H'' of ''A'' by ''H'' is 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 ...
''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 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 = \bigoplus_ A_\omega as the base of the wreath product. In this case the elements of ''K'' are sequences (''a''ω) of elements in ''A'' indexed by Ω of which all but finitely many ''a''ω are the
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''≀Ω''H'' may stand for the unrestricted wreath product ''A'' WrΩ ''H'' or the restricted wreath product ''A'' wrΩ ''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 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''''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'' 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 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'' WrΩ ''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'' WrΩ ''H'' (and therefore also ''A'' wrΩ ''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 ΛΩ. *: An element in ΛΩ is a sequence (''λ''''ω'') indexed by the ''H''-set Ω. Given an element its operation on (''λ''''ω'') ∈ ΛΩ 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 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 :: 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 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''''n'' of degree ''n'' is given by :: ''φ''(''σ'')(α1,..., ''α''''n'') := (''α''''σ''(1),..., ''α''''σ''(''n'')). * ''S''2≀''S''''n'' (
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''''n'' on is as above. Since the symmetric group ''S''2 of degree 2 is
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 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 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''''p''''n''. Then ''P'' is
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''''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
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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''2 ≀ ''S''2 ≀ ''...'' ≀ ''S''2 is the automorphism group of a complete
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binary tree
.


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