Wreath product

Summary

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

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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

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The structure of the wreath product of   by   depends on the  -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,   may stand for the unrestricted wreath product   or the restricted wreath product  .
  • In literature, the  -set   may be omitted from the notation even if  .
  • In the special case that   is the symmetric group of degree  , it is common in the literature to assume that   (with the natural action of  ) and then omit   from the notation. That is,   commonly denotes   instead of the regular wreath product  . In the first case the base group is the product of   copies of  , in the latter it is the product of n! copies of  .

Properties

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Agreement of unrestricted and restricted wreath product on finite Ω

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Since the finite direct product is the same as the finite direct sum of groups, it follows that the unrestricted wreath product   and the restricted wreath product   are equal if   is finite. In particular, this is true when   and   is finite.

Subgroup

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  is always a subgroup of  .

Cardinality

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If  ,   and   are finite, then

 .[2]

Universal embedding theorem

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If   is an extension of   by  , then there exists a subgroup of the unrestricted wreath product   which is isomorphic to  .[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

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If the group   acts on a set   then there are two canonical ways to construct sets from   and   on which   (and therefore also  ) can act.

  • The imprimitive wreath product action on  :
    If   and  , then
     
  • The primitive wreath product action on  :
    An element in   is a sequence   indexed by the  -set  . Given an element  , its operation on   is given by
     

Examples

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  • The lamplighter group is the restricted wreath product  .
  • The generalized symmetric group is  . The base of this wreath product is the  -fold direct product  , where the action   of the symmetric group   is given by  .[5]
    • As a special case, we have the hyperoctahedral group   (since   is isomorphic to  ).[6]
  • The smallest non-trivial wreath product is  , which is the two-dimensional case of the above hyperoctahedral group. It is the symmetry group of the square, also called  , the dihedral group of order 8.
  • Let   be a prime and let  . Let   be a Sylow p-subgroup of the symmetric group  . Then   is isomorphic to the iterated regular wreath product   of   copies of  . Here   and   for all  .[7][8] For instance, the Sylow 2-subgroup of   is the above   group.
  • The Rubik's Cube group is a normal subgroup of index 12 in the product of wreath products,  , the factors corresponding to the symmetries of the 8 corners and 12 edges.
  • The Sudoku validity-preserving transformations (VPT) group contains the double wreath product  , where the factors are the permutation of rows/columns within a 3-row or 3-column band or stack ( ), the permutation of the bands/stacks themselves ( ) and the transposition, which interchanges the bands and stacks ( ). Here, the two index sets   are firstly the set of bands (resp. stacks), so  , and secondly the set {bands, stacks} (so  ). Accordingly,   and  .
  • Wreath products arise naturally in the symmetries of complete rooted trees and their graphs. For example, the repeated (iterated) wreath product   is the automorphism group of a complete binary tree.

References

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  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)
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