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.
Definition.
Let "A" and "H" be groups and Ω a set with "H" acting on it (from the left). Let "K" be the direct product
of copies of "A"ω := "A" indexed by the set Ω. The elements of "K" can be seen as arbitrary sequences ("a"ω) of elements of "A" indexed by Ω with component-wise multiplication. Then the action of "H" on Ω extends in a natural way to an action of "H" on the group "K" by
Then the unrestricted wreath product "A" WrΩ "H" of "A" by "H" is the semidirect product "K" ⋊ "H". The subgroup "K" of "A" WrΩ "H" is called the base of the wreath product.
The restricted wreath product "A" wrΩ "H" 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 elements of "K" are sequences ("a"ω) of elements in "A" indexed by Ω of which all but finitely many "a"ω are the identity element of "A".
In the most common case, one takes Ω := "H", where "H" acts in a natural way on itself by left multiplication. In this case, the unrestricted and restricted wreath product may be denoted by "A" Wr "H" and "A" wr "H" respectively. This is called the regular wreath product.
Notation and conventions.
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.
Properties.
Agreement of unrestricted and restricted wreath product on finite Ω.
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 the "H"-set Ω is finite. In particular this is true when Ω = "H" is finite.
Subgroup.
"A" wrΩ "H" is always a subgroup of "A" WrΩ "H".
Cardinality.
If "A", "H" and Ω are finite, then
Universal embedding theorem.
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". This is also known as the "Krasner–Kaloujnine embedding theorem". The Krohn–Rhodes theorem involves what is basically the semigroup equivalent of this.
Canonical actions of wreath products.
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.