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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 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'' 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 ''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'' 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 (''a''ω) of elements in ''A'' indexed by Ω of which all but finitely many ''a''ω 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「wreath product」の詳細全文を読む スポンサード リンク
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