Wreath product

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 \overline{a} = (a_{\omega})_{\omega \in \Omega} 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

: h \cdot (a_{\omega}){\omega \in \Omega} := (a{h^{-1} \cdot \omega})_{\omega \in \Omega}

for all h \in H and all (a_{\omega})_{\omega \in \Omega} \in A^{\Omega}.

Then the unrestricted wreath product A \text{ Wr}_{\Omega} H of A by H is the semidirect product A^{\Omega} \rtimes H with the action of H on A^{\Omega} given above. The subgroup A^{\Omega} of A^{\Omega} \rtimes H is called the base of the wreath product.

The restricted wreath product A \text{ wr}_{\Omega} H 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 A \text{ Wr } H and A \text{ 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\wr_\Omega H may stand for the unrestricted wreath product A\operatorname{Wr}\Omega H or the restricted wreath product A\operatorname{wr}\Omega H.
• In literature, the H-set \Omega may be omitted from the notation even if \Omega\neq H.
• In the special case that H=S_n is the symmetric group of degree n, it is common in the literature to assume that \Omega={1,\dots,n} (with the natural action of S_n) and then omit \Omega from the notation. That is, A\wr S_n commonly denotes A\wr_{{1,\dots,n}} S_n instead of the regular wreath product A\wr_{S_n} S_n. 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 wreath product A\operatorname{Wr}\Omega H and the restricted wreath product A\operatorname{wr}\Omega H are equal if \Omega is finite. In particular, this is true when \Omega = H and H is finite.

Subgroup

A\operatorname{wr}\Omega H is always a subgroup of A\operatorname{Wr}\Omega H.

Cardinality

If A, H and \Omega are finite, then :|A\wr_\Omega! H| = |A|^{|\Omega|}|H|.

Universal embedding theorem

Main article: Universal embedding theorem

If G is an extension of A by H, then there exists a subgroup of the unrestricted wreath product A\wr 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 \Lambda then there are two canonical ways to construct sets from \Omega and \Lambda on which A\operatorname{Wr}\Omega H (and therefore also A\operatorname{wr}\Omega H) can act.

• The imprimitive wreath product action on \Lambda\times\Omega:
• : If ((a_\omega), h)\in A \operatorname{Wr}_\Omega H and (\lambda,\omega')\in \Lambda\times \Omega, then
• :: ((a_\omega), h) \cdot (\lambda,\omega') := (a_{h(\omega')}\lambda, h\omega').
• The primitive wreath product action on \Lambda^\Omega:
• : An element in \Lambda^\Omega is a sequence (\lambda_\omega) indexed by the H-set \Omega. Given an element ((a_\omega), h)\in A \operatorname{Wr}\Omega H, its operation on (\lambda\omega) \in \Lambda^\Omega is given by
• :: ((a_\omega), h) \cdot (\lambda_\omega) := (a_{h^{-1}\omega}\lambda_{h^{-1}\omega}).

Examples

• The lamplighter group is the restricted wreath product C_2 \wr \mathbb{Z}.

• The generalized symmetric group is C_m \wr S_n. The base of this wreath product is the n-fold direct product C_m^n = C_m\times\cdots\times C_m, where the action \phi:S_n \to \text{Aut}(C_m^n) of the symmetric group S_n is given by \phi(\sigma)(\alpha_1,\dots,\alpha_n)=(\alpha_{\sigma(1)},\dots,\alpha_{\sigma(n)}).

  • As a special case, we have the hyperoctahedral group S_2 \wr S_n (since S_2 is isomorphic to C_2).

• The smallest non-trivial wreath product is C_2 \wr C_2, which is the two-dimensional case of the above hyperoctahedral group. It is the symmetry group of the square, also called D_8, the dihedral group of order 8.

• Let p be a prime and let n \geq 1. Let P be a Sylow p-subgroup of the symmetric group S_{p^n}. Then P is isomorphic to the iterated regular wreath product W_n = C_p \wr \cdots \wr C_p of n copies of C_p. Here W_1=C_p and W_k=W_{k - 1} \wr C_p for all k \geq 2. For instance, the Sylow 2-subgroup of S_4 is the above C_2 \wr C_2 group.

• The Rubik's Cube group is a normal subgroup of index 12 in the product of wreath products, (C_3 \wr S_8) \times (C_2 \wr S_{12}), 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 (S_3\wr S_3)\wr S_2, where the factors are the permutation of rows/columns within a 3-row or 3-column band or stack (S_3), the permutation of the bands/stacks themselves (S_3) and the transposition, which interchanges the bands and stacks (S_2). Here, the two index sets \Omega_1,\Omega_2 are firstly the set of bands (resp. stacks), so |\Omega_1|=3, and secondly the set {bands, stacks} (so |\Omega_2|=2). Accordingly, |S_3\wr S_3|=|S_3|^3|S_3|=6^4 and |(S_3\wr S_3)\wr S_2|=|S_3\wr S_3|^2|S_2|=6^8\times 2.

• Wreath products arise naturally in the symmetries of complete rooted trees and their graphs. For example, the repeated (iterated) wreath product S_2\wr S_2\wr\cdots\wr S_2 is the automorphism group of a complete binary tree.

References

References

1. (1998). "Wreath products". Springer.
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].
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)