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 groups and also provide a way of constructing interesting examples of groups.
Given two groups and there exists two variations of the wreath product: the unrestricted wreath product (also written ) and the restricted wreath product . Given a set with an -action there exists a generalisation of the wreath product which is denoted by or respectively.
of copies of indexed by the set . The elements of can be seen as arbitrary sequences of elements of indexed by with component wise multiplication. Then the action of on extends in a natural way to an action of on the group by
Then the unrestricted wreath product of by is the semidirect product . The subgroup of is called the base of the wreath product.
The restricted wreath product 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 are sequences of elements in indexed by of which all but finitely many are the identity element of .
The group acts in a natural way on itself by left multiplication. Thus we can choose . In this special (but very common) case the unrestricted and restricted wreath product may be denoted by and respectively. We say in this case that the wreath product is regular.
- Since the finite direct product is the same as the finite direct sum of groups it follows that the unrestricted and the restricted wreath product agree if is finite. In particluar this is true when is finite.
- Universal Embedding Theorem: If is an extension of by , then there exists a subgroup of the unrestricted wreath product which is isomorphic to .
- If , and are finite, then
- From this follows immediately that in general the wreath product can neither be associative nor commutative.
- For example, for finite groups , and it is easy to construct examples where
- and .
Canonical Actions of Wreath Products
If the group acts on a set then there are two canonical ways to construct sets from and on which can act.
- The imprimitive wreath product action on .
- The primitive wreath product action on . An element in is a sequence indexed by the -set . Given an element its operation on is given by
- The restricted wreath product is known as the Lamplighter group.
- The base of this wreath product is the -fold direct product
- of copies of where the action of symmetric group of degree n is given by
- The action of on is as above. Since the symmetric group of degree 2 is isomorphic to the hyperoctahedral group is a special case of a geleralized symmetric group.
- (Kaloujnine, 1948) Let be a prime and let . Let be a Sylow -subgroup of the symmetric group of degree . Then is isomorphic to the iterated regular wreath product of copies of . Here and for all .
- Joseph J. Rotman, An Introduction to the Theory of Groups, pp. 172ff. (1995)
- 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)
- 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. 615-620
- 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.
- 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)
- Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995)