Group formed from the action of one group on many copies of another, based on the semidirect product
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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[1]), there exist two variants 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 variants coincide when , , and are all finite. Either variant is also denoted as (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.
^Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1998), "Wreath products", Notes on Infinite Permutation Groups, Lecture Notes in Mathematics, vol. 1698, Berlin, Heidelberg: Springer, pp. 67–76, doi:10.1007/bfb0092558, ISBN 978-3-540-49813-1, retrieved 2021-05-12
In group theory, the wreathproduct is a special combination of two groups based on the semidirect product. It is formed by the action of one group on...
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semidirect product is also solvable. It is also closed under wreathproduct: If G and H are solvable, and X is a G-set, then the wreathproduct of G and...
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\varphi } . ≀ In group theory, G ≀ H {\displaystyle G\wr H} denotes the wreathproduct of the groups G and H. It is also denoted as G wr H {\displaystyle...
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is nonempty and closed under products and inverses. Closed under products means that for every a and b in H, the product ab is in H. Closed under inverses...
Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreathproduct simple finite infinite continuous multiplicative...
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