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 "A" and "H" (sometimes known as the "bottom" and "top"), there exist two variations of the wreath product: the unrestricted wreath product "A" Wr "H" and the restricted wreath product "A" wr "H". The general form, denoted by "A" WrΩ "H" or "A" wrΩ "H" respectively, uses a set "Ω" with an "H"-action; when unspecified, usually "Ω" = "H" (a regular wreath product), though a different "Ω" is sometimes implied. The two variations coincide when "A", "H", and "Ω" are all finite. Either variation is also denoted as formula_1 (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.
Q874429
group theory
Concept
Q83478
groups
Concept
Q291126
semidirect product
Concept
Q288465
action
Concept
Q33456
exponentiation
Concept
Q1412905
permutation group
Concept
Q36161
set
Concept
Q288465
"H"-action
Concept
Q8819
Unicode
MISC
Q207348
semigroup
Concept
Q6438788
Krohn–Rhodes structure theory
UNK
Q1786640
Wreath product
Concept
+1^^http://www.w3.org/2001/XMLSchema#decimal
1
NUMBER
++2240^^http://www.w3.org/2001/XMLSchema#decimal
+2240
NUMBER
subclass of
studies
facet of