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.
(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
(a regular wreath product), though a different
(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.
be a group acting on a set
, with a group operation given by pointwise multiplication.
Then the unrestricted wreath product
is called the base of the wreath product.
The restricted wreath product
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 base consists of all sequences in
with finitely many non-identity entries.
acts on itself by left multiplication.
In this case, the unrestricted and restricted wreath product may be denoted by
This is called the regular wreath product.
The structure of the wreath product of A by H depends on the H-set Ω and in case Ω is infinite it also depends on whether one uses the restricted or unrestricted wreath product.
However, in literature the notation used may be deficient and one needs to pay attention to the circumstances.
Since the finite direct product is the same as the finite direct sum of groups, it follows that the unrestricted A WrΩ H and the restricted wreath product A wrΩ H agree if Ω is finite.
In particular this is true when Ω = H and H is finite.
A wrΩ H is always a subgroup of A WrΩ H. If A, H and Ω are finite, then Universal embedding theorem: If G is an extension of A by H, then there exists a subgroup of the unrestricted wreath product A≀H which is isomorphic to G.[3] This is also known as the Krasner–Kaloujnine embedding theorem.
The Krohn–Rhodes theorem involves what is basically the semigroup equivalent of this.
[4] If the group A acts on a set Λ then there are two canonical ways to construct sets from Ω and Λ on which A WrΩ H (and therefore also A wrΩ H) can act.