Wreath product

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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 A and H there exists 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.

Contents

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 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.

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 AΩH may stand for the unrestricted wreath product A WrΩ H or the restricted wreath product A wrΩ H.
  • Similarly, AH may stand for the unrestricted regular wreath product A Wr H or the restricted regular wreath product A wr H.

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