Wreath product
Algebraic structure → Group theory Group theory 


Modular groups

Infinite dimensional Lie group

In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. 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, 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 Haction there exists a generalisation of the wreath product which is denoted by A Wr_{Ω} H or A wr_{Ω} H respectively.
The notion generalizes to semigroups and is a central construction in the KrohnRhodes structure theory of finite semigroups.
Definition
Let A and H be groups and Ω a set with H acting on it. Let K be the direct product
of copies of A_{ω} := A indexed by the set Ω. The elements of K can be seen as arbitrary sequences (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
 .
Then the unrestricted wreath product A Wr_{Ω} H of A by H is the semidirect product K ⋊ H. The subgroup K of A Wr_{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 direct sum
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 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 Hset 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≀_{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 Hset 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={1,...,n} (with the natural action of S_{n}) and then omit Ω from the notation. That is, A≀S_{n} commonly denotes A≀_{{1,...,n}}S_{n} instead of the regular wreath product A≀_{Sn}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
 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 Hset Ω is finite. In particular this is true when Ω = H is finite.
 A wr_{Ω} H is always a subgroup of A Wr_{Ω} H.
 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.^{[1]} This is also known as the KrasnerKaloujnine embedding theorem. The KrohnRhodes theorem involves what is basically the semigroup equivalent of this.^{[2]}
 If A, H and Ω are finite, then
 A≀_{Ω}H = A^{Ω}H.^{[3]}
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 ((a_{ω}),h)∈A Wr_{Ω} H and (λ,ω')∈Λ×Ω, then
 .
 The primitive wreath product action on Λ^{Ω}.
 An element in Λ^{Ω} is a sequence (λ_{ω}) indexed by the Hset Ω. Given an element ((a_{ω}), h) ∈ A Wr_{Ω} H its operation on (λ_{ω})∈Λ^{Ω} is given by
 .
Examples
 The Lamplighter group is the restricted wreath product ℤ_{2}≀ℤ.
 ℤ_{m}≀S_{n} (Generalized symmetric group).
 The base of this wreath product is the nfold 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)(α_{1},..., α_{n}) := (α_{σ(1)},..., α_{σ(n)}).^{[4]}
 S_{2}≀S_{n} (Hyperoctahedral group).
 The action of S_{n} on {1,...,n} is as above. Since the symmetric group S_{2} of degree 2 is isomorphic to ℤ_{2} the hyperoctahedral group is a special case of a generalized symmetric group.^{[5]}
 The smallest nontrivial wreath product is ℤ_{2}≀ℤ_{2}, which is the twodimensional 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 psubgroup of the symmetric group S_{pn} of degree 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_{k1}≀ℤ_{p} for all k≥2.^{[6]}^{[7]} For instance, the Sylow 2subgroup 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.
References
 ↑ M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de groupes III", Acta Sci. Math. Szeged 14, pp. 6982 (1951)
 ↑ J D P Meldrum (1995). Wreath Products of Groups and Semigroups. Longman [UK] / Wiley [US]. p. ix. ISBN 9780582026933.
 ↑ Joseph J. Rotman, An Introduction to the Theory of Groups, p. 172 (1995)
 ↑ J. W. Davies and A. O. Morris, "The Schur Multiplier of the Generalized Symmetric Group", J. London Math. Soc (2), 8, (1974), pp. 615620
 ↑ P. Graczyk, G. Letac and H. Massam, "The Hyperoctahedral Group, Symmetric Group Representations and the Moments of the Real Wishart Distribution", J. Theoret. Probab. 18 (2005), no. 1, 142.
 ↑ Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995)
 ↑ L. Kaloujnine, "La structure des pgroupes de Sylow des groupes symétriques finis", Annales Scientifiques de l'École Normale Supérieure. Troisième Série 65, pp. 239–276 (1948)