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 A≀H) 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 Wr_Ω H or A wr_Ω 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 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.

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