Editing Wreath product (section)

== Examples ==

* The [[lamplighter group]] is the restricted wreath product <math>\mathbb{Z}_2 \wr \mathbb{Z}</math>.

* <math>\mathbb{Z}_m \wr S_n</math>(the [[generalized symmetric group]]). The base of this wreath product is the ''n''-fold direct product <math>\mathbb{Z}_m^n = \mathbb{Z}_m ... \mathbb{Z}_m</math>of copies of <math>\mathbb{Z}_m</math> where the action <math>\phi:S_n \to \text{Aut}(\mathbb{Z}_m^n)</math> of the [[symmetric group]] ''S''<sub>''n''</sub> of degree ''n'' is given by ''φ''(''σ'')(α<sub>1</sub>,..., ''α''<sub>''n''</sub>) := (''α''<sub>''σ''(1)</sub>,..., ''α''<sub>''σ''(''n'')</sub>).<ref>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</ref>

* <math>S_2 \wr S_n</math>(the [[hyperoctahedral group]]).
* The action of ''S''<sub>''n''</sub> on {1,...,''n''} is as above. Since the symmetric group ''S''<sub>2</sub> of degree 2 is [[Group isomorphism|isomorphic]] to <math>\mathbb{Z}_2</math> the hyperoctahedral group is a special case of a generalized symmetric group.<ref>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.</ref>

* The smallest non-trivial wreath product is <math>\mathbb{Z}_2 \wr \mathbb{Z}_2</math>, which is the two-dimensional case of the above hyperoctahedral group. It is the symmetry group of the square, also called ''D''<sub>4</sub>, the [[dihedral group]] of order 8.
* Let ''p'' be a [[Prime number|prime]] and let <math>n \geq 1</math>. Let ''P'' be a [[Sylow theorems|Sylow ''p''-subgroup]] of the symmetric group ''S''<sub>''p''<sup>''n''</sup></sub>. Then ''P'' is [[Group isomorphism|isomorphic]] to the iterated regular wreath product <math>W_n = \mathbb{Z}_p \wr ... \wr \mathbb{Z}_p</math> of ''n'' copies of <math>\mathbb{Z}_p</math>. Here <math>W_1:=\mathbb{Z}_p</math> and <math>W_k:=W_{k - 1} \wr \mathbb{Z}_p</math> for all <math>k \geq 2</math>.<ref>Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995)</ref><ref>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)</ref> For instance, the Sylow 2-subgroup of S<sub>4</sub> is the above <math>\mathbb{Z}_2 \wr \mathbb{Z}_2</math> group.

* The [[Rubik's Cube group]] is a normal subgroup of index 12 in the product of wreath products, <math>(\mathbb{Z}_3 \wr S_8) \times (\mathbb{Z}_2 \wr S_{12})</math>, the factors corresponding to the symmetries of the 8 corners and 12 edges.

* The [[Mathematics of Sudoku#The sudoku symmetry group|Sudoku validity-preserving transformations (VPT) group]] contains the double wreath product (''S''<sub>3</sub> ≀ ''S''<sub>3</sub>) ≀ ''S''<sub>2</sub>, where the factors are the permutation of rows/columns within a 3-row or 3-column ''band'' or ''stack'' (''S''<sub>3</sub>), the permutation of the bands/stacks themselves (''S''<sub>3</sub>) and the transposition, which interchanges the bands and stacks (''S''<sub>2</sub>). Here, the index sets ''Ω'' are the set of bands (resp. stacks) (|''Ω''| = 3) and the set {bands, stacks} (|''Ω''| = 2). Accordingly, |''S''<sub>3</sub> ≀ ''S''<sub>3</sub>| = |''S''<sub>3</sub>|<sup>3</sup>|''S''<sub>3</sub>| = (3!)<sup>4</sup> and |(''S''<sub>3</sub> ≀ ''S''<sub>3</sub>) ≀ ''S''<sub>2</sub>| = |''S''<sub>3</sub> ≀ ''S''<sub>3</sub>|<sup>2</sup>|''S''<sub>2</sub>| = (3!)<sup>8</sup> × 2.

*Wreath products arise naturally in the symmetries of complete rooted [[Tree (data structure)|trees]] and their [[Graph (discrete mathematics)|graphs]]. For example, the repeated (iterated) wreath product ''S''<sub>2</sub> ≀ ''S''<sub>2</sub> ≀ ''...'' ≀ ''S''<sub>2</sub> is the automorphism group of a complete [[binary tree]].