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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 and (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 or respectively, requires that acts on some set ; when unspecified, usually (a regular wreath product), though a different is sometimes implied. The two variants coincide when , , and are all finite. Either variant is also denoted as (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 be a group and let be a group acting on a set (on the left). The direct product of with itself indexed by is the set of sequences in , indexed by , with a group operation given by pointwise multiplication. The action of on can be extended to an action on by reindexing, namely by defining

for all and all .

Then the unrestricted wreath product of by is the semidirect product with the action of on given above. The subgroup 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 with finitely many non-identity entries. The two definitions coincide when is finite.

In the most common case, , and 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

|AΩH| = |A||Ω||H|.[2]

Universal embedding theorem

Main article: 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 AH 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]

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

References

  1. ^ Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1998), "Wreath products", Notes on Infinite Permutation Groups, Lecture Notes in Mathematics, vol. 1698, Berlin, Heidelberg: Springer, pp. 67–76, doi:10.1007/bfb0092558, ISBN 978-3-540-49813-1, retrieved 2021-05-12
  2. ^ Joseph J. Rotman, An Introduction to the Theory of Groups, p. 172 (1995)
  3. ^ M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de groupes III", Acta Sci. Math. 14, pp. 69–82 (1951)
  4. ^ J D P Meldrum (1995). Wreath Products of Groups and Semigroups. Longman [UK] / Wiley [US]. p. ix. ISBN 978-0-582-02693-3.
  5. ^ J. W. Davies and A. O. Morris, "The Schur Multiplier of the Generalized Symmetric Group", J. London Math. Soc. (2), 8, (1974), pp. 615–620
  6. ^ P. Graczyk, G. Letac and H. Massam, "The Hyperoctahedral Group, Symmetric Group Representations and the Moments of the Real Wishart Distribution", J. Theoret. Probab. 18 (2005), no. 1, 1–42.
  7. ^ Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995)
  8. ^ L. Kaloujnine, "La structure des p-groupes de Sylow des groupes symétriques finis", Annales Scientifiques de l'École Normale Supérieure. Troisième Série 65, pp. 239–276 (1948)