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 variants of the wreath product: the unrestricted wreath product**** and the restricted wreath product. The general form, denoted by or respectively, requires that H acts on some set \Omega; when unspecified, usually \Omega = H (a regular wreath product), though a different \Omega is sometimes implied. The two variants coincide when A, H, and \Omega are all finite. Either variant is also denoted as A \wr H (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.

Definition

Let A be a group and let H be a group acting on a set \Omega (on the left). The direct product A^{\Omega} of A with itself indexed by \Omega is the set of sequences in A, indexed by \Omega, with a group operation given by pointwise multiplication. The action of H on \Omega can be extended to an action on A^{\Omega} by reindexing, namely by defining for all h \in H and all. Then the unrestricted wreath product of A by H is the semidirect product with the action of H on A^{\Omega} given above. The subgroup A^{\Omega} of 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 A^{\Omega} with finitely many non-identity entries. The two definitions coincide when \Omega is finite. In the most common case, \Omega = H, and H acts on itself by left multiplication. In this case, the unrestricted and restricted wreath product may be denoted by and 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 Ω is finite. In particular this is true when Ω = H and 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.

Examples