In group theory, the wreath product is a specialized product of two groups, based on a semidirect product. 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, there exist two variations of the wreath product: the unrestricted wreath productA Wr H and the restricted wreath productA wr H. Given a set Ω with an H-action there exists a generalization of the wreath product which is denoted by A WrΩH or A wrΩH respectively. 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 Hacting on it. Let K be the direct product of copies of Aω := A indexed by the set Ω. The elements of K can be seen as arbitrary sequences 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 productA WrΩH of A by H is the semidirect product K ⋊ H. The subgroupK of A WrΩH is called the base of the wreath product. The restricted wreath productA wrΩH 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 of elements in A indexed by Ω of which all but finitely manyaω are the identity element ofA. 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.
In literature A≀ΩH may stand for the unrestricted wreath product A WrΩH or the restricted wreath product A wrΩH.
Similarly, A≀H may stand for the unrestricted regular wreath product A Wr H or the restricted regular wreath product A wr H.
In literature the H-set Ω may be omitted from the notation even if Ω ≠ H.
In the special case that H = Sn is the symmetric group of degree n it is common in the literature to assume that Ω = and then omit Ω from the notation. That is, A≀Sn commonly denotes A≀Sn instead of the regular wreath product A≀SnSn. In the first case the base group is the product of n copies of A, in the latter it is the product of n! copies of A.
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 WrΩH and the restricted wreath product A wrΩH agree if the H-set Ω is finite. In particular this is true when Ω = H is finite.
Subgroup
A wrΩH is always a subgroup of A WrΩH.
Cardinality properties
If A, H and Ω are finite, then
Universal embedding theorem
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 WrΩH can act.
Let p be a prime and let n≥1. Let P be a Sylow p-subgroup of the symmetric group Spn of degree pn. Then P is isomorphic to the iterated regular wreath product Wn = ℤp ≀ ℤp≀...≀ℤp of n copies of ℤp. Here W1 := ℤp and Wk := Wk−1≀ℤp for all k ≥ 2. For instance, the Sylow 2-subgroup of S4 is the above ℤ2≀ℤ2 group.
The Rubik's Cube group is a subgroup of index 12 in the product of wreath products, ×, the factors corresponding to the symmetries of the 8 corners and 12 edges.
