Wreath product
Encyclopedia
In mathematics
, the wreath product of group theory
is a specialized product of two groups, based on a semidirect product
. Wreath products are an important tool in the classification of permutation group
s and also provide a way of constructing interesting examples of groups.
Given two groups A and H there exist two variations of the wreath product: the unrestricted wreath product A Wr H (also written A≀H) and the restricted wreath product A wr H. Given a set Ω with an H-action
there exists a generalisation of the wreath product which is denoted by A WrΩ H or A wrΩ H respectively.
on it. Let K be the direct product
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
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
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.
The group H acts
in a natural way on itself by left multiplication. Thus we can choose Ω := H. In this special (but very common) case the unrestricted and restricted wreath product may be denoted by A Wr H and A wr H respectively. We say in this case that the wreath product is regular.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the wreath product of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
is a specialized product of two groups, based on a semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
. Wreath products are an important tool in the classification of permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
s and also provide a way of constructing interesting examples of groups.
Given two groups A and H there exist two variations of the wreath product: the unrestricted wreath product A Wr H (also written A≀H) and the restricted wreath product A wr H. Given a set Ω with an H-action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
there exists a generalisation of the wreath product which is denoted by A WrΩ H or A wrΩ H respectively.
Definition
Let A and H be groups and Ω a set with H actingGroup action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on it. Let K be the direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
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
-
.
Then the unrestricted wreath product A WrΩ H of A by H is the semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct 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
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
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
of A.
The group H acts
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
in a natural way on itself by left multiplication. Thus we can choose Ω := H. In this special (but very common) case the unrestricted and restricted wreath product may be denoted by A Wr H and A wr H respectively. We say in this case that the wreath product is regular.
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 on the circumstances.- In literature A≀ΩH 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 = Sn is the symmetric groupSymmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
of degree n it is common in literature to assume that Ω={1,...,n} (with the natural action of Sn) and then omit Ω from the notation. That is, in literature A≀Sn commonly denotes A≀{1,...,n}Sn instead of the regular wreath product A≀SnSn. In the first case the base group is the direct sum of n copies of A, in the latter it is the direct sum of n!FactorialIn mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
copies of A.
Properties
- 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.
- A wrΩ H is always a subgroupSubgroupIn group theory, given a 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, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of A WrΩ H.
- Universal Embedding Theorem: If G is an extensionGroup extensionIn mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
of A by H, then there exists a subgroup of the unrestricted wreath product A≀H which is isomorphic to G.
- If A, H and Ω are finite, then
-
- |A≀ΩH| = |A||Ω||H|.
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
-
-
.
-
- 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
-
-
.
-
Examples
- The Lamplighter group is the restricted wreath product ℤ2≀ℤ.
- ℤm≀Sn (Generalized symmetric groupGeneralized symmetric groupIn mathematics, the generalized symmetric group is the wreath product S := Z_m \wr S_n of the cyclic group of order m and the symmetric group on n letters.- Examples :...
).
- The base of this wreath product is the n-fold direct product
-
- ℤmn = ℤm × ... × ℤm
- of copies of ℤm where the action φ : Sn → Aut(ℤmn) of the symmetric groupSymmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
Sn of degree n is given by
-
- φ(σ)(α1,..., αn) := (ασ(1),..., ασ(n)).
- S2≀Sn (Hyperoctahedral groupHyperoctahedral groupIn mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. Groups of this type are identified by a parameter n, the dimension of the hypercube....
).
- The action of Sn on {1,...,n} is as above. Since the symmetric group S2 of degree 2 is isomorphicGroup isomorphismIn abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
to ℤ2 the hyperoctahedral group is a special case of a generalized symmetric group.
- Let p be a primePrime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
and let n≤1. Let P be a Sylow p-subgroup of the symmetric group Spn of degree pn. Then P is isomorphicGroup isomorphismIn abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called 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.
- The Rubik's cube groupRubik's Cube groupThe Rubik's Cube group is a mathematical group which corresponds to the set of all cube operations on Rubik's Cube, with function composition as the group operation....
is a subgroup of small index in the product of wreath products, (ℤ3≀S8) × (ℤ2≀S12), the factors corresponding to the symmetries of the 8 corners and 12 edges.