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.

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

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