Wreath product

In group theory, the wreath product 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. In group theory, the wreath product 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 generalization 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.