Wreath product

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 AH) 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.

The notion generalizes to semigroups and is a central construction in the Krohn-Rhodes structure theory of finite semigroups.


Let A and H be groups and Ω a set with H acting 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.

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 on the circumstances.


|AΩH| = |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 WrΩ H (and therefore also A wrΩ H) can act.

If ((aω),h)∈A WrΩ H and (λ,ω')∈Λ×Ω, then
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


The base of this wreath product is the n-fold direct product
mn = ℤm × ... × ℤm
of copies of ℤm where the action φ : Sn → Aut(ℤmn) of the symmetric group Sn of degree n is given by
φ(σ)(α1,..., αn) := (ασ(1),..., ασ(n)).[4]
The action of Sn on {1,...,n} is as above. Since the symmetric group S2 of degree 2 is isomorphic to ℤ2 the hyperoctahedral group is a special case of a generalized symmetric group.[5]


  1. 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. 69-82 (1951)
  2. J D P Meldrum (1995). Wreath Products of Groups and Semigroups. Longman [UK] / Wiley [US]. p. ix. ISBN 978-0-582-02693-3.
  3. Joseph J. Rotman, An Introduction to the Theory of Groups, p. 172 (1995)
  4. J. W. Davies and A. O. Morris, "The Schur Multiplier of the Generalized Symmetric Group", J. London Math. Soc (2), 8, (1974), pp. 615-620
  5. 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, 1-42.
  6. Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995)
  7. 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)

External links

This article is issued from Wikipedia - version of the 3/9/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.