Permutation groups | Binary operations | Group products

Wreath product

In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used in the classification of permutation groups and also provide a way of constructing interesting examples of groups. Given two groups and (sometimes known as the bottom and top), there exist two variations of the wreath product: the unrestricted wreath product and the restricted wreath product . The general form, denoted by or respectively, requires that acts on some set ; when unspecified, usually (a regular wreath product), though a different is sometimes implied. The two variations coincide when , , and are all finite. Either variation is also denoted as (with \wr for the LaTeX symbol) or A ≀ H (Unicode U+2240). The notion generalizes to semigroups and is a central construction in the Krohn–Rhodes structure theory of finite semigroups. (Wikipedia).

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Exponentiation | Graph (discrete mathematics) | Group extension | Lamplighter group | Hyperoctahedral group | Group (mathematics) | Identity element | Permutation group | Sylow theorems | Symmetric group | Group isomorphism | Factorial | Semigroup | Direct product of groups | Generalized symmetric group | Dihedral group | Semidirect product | Direct sum of groups | Group theory | Prime number | Tree (data structure) | Subgroup | Mathematics of Sudoku | Rubik's Cube group | Binary tree | Universal embedding theorem