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 A {\displaystyle A} and H {\displaystyle H} (sometimes known as the bottom and top), there exist two variants of the wreath product: the unrestricted wreath product A Wr H {\displaystyle A{\text{ Wr }}H} and the restricted wreath product A wr H {\displaystyle A{\text{ wr }}H} . The general form, denoted by A Wr Ω H {\displaystyle A{\text{ Wr}}_{\Omega }H} or A wr Ω H {\displaystyle A{\text{ wr}}_{\Omega }H} respectively, requires that H {\displaystyle H} acts on some set Ω {\displaystyle \Omega } ; when unspecified, usually Ω = H {\displaystyle \Omega =H} (a regular wreath product), though a different Ω {\displaystyle \Omega } is sometimes implied. The two variants coincide when A {\displaystyle A} , H {\displaystyle H} , and Ω {\displaystyle \Omega } are all finite. Either variant is also denoted as A ≀ H {\displaystyle A\wr H} (with \wr for the LaTeX symbol) or A ≀ H (Unicode U+2240). The notion generalizes to semigroups and, as such, is a central construction in the Krohn–Rhodes structure theory of finite semigroups.
Definition Let A {\displaystyle A} be a group and let H {\displaystyle H} be a group acting on a set Ω {\displaystyle \Omega } (on the left). The direct product A Ω {\displaystyle A^{\Omega }} of A {\displaystyle A} with itself indexed by Ω {\displaystyle \Omega } is the set of sequences a ¯ = ( a ω ) ω ∈ Ω {\displaystyle {\overline {a}}=(a_{\omega })_{\omega \in \Omega }} in A {\displaystyle A} , indexed by Ω {\displaystyle \Omega } , with a group operation given by pointwise multiplication. The action of H {\displaystyle H} on Ω {\displaystyle \Omega } can be extended to an action on A Ω {\displaystyle A^{\Omega }} by reindexing, namely by defining
h ⋅ ( a ω ) ω ∈ Ω := ( a h − 1 ⋅ ω ) ω ∈ Ω {\displaystyle h\cdot (a_{\omega })_{\omega \in \Omega }:=(a_{h^{-1}\cdot \omega })_{\omega \in \Omega }}
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