Wreath product

Last updated

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.

Contents

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

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

Related Research Articles

<span class="mw-page-title-main">Symmetric group</span> Type of group in abstract algebra

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .

<span class="mw-page-title-main">Lagrange's theorem (group theory)</span> The order of a subgroup of a finite group G divides the order of G

In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then |H| is a divisor of |G|, i.e. the order of every subgroup H divides the order of group G.

<span class="mw-page-title-main">Generating set of a group</span> Abstract algebra concept

In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination of finitely many elements of the subset and their inverses.

<span class="mw-page-title-main">Solvable group</span> Group with subnormal series where all factors are abelian

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

In mathematics, a symplectic matrix is a matrix with real entries that satisfies the condition

In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted or or . Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula

In mathematics, a generalized permutation matrix is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is

<span class="mw-page-title-main">Modular group</span> Orientation-preserving mapping class group of the torus

In mathematics, the modular group is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.

In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids.

In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups, generalizing the notion of a single coset.

In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space obtained from the action of on by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation of the symmetric group over the complex numbers. A similar construction works over any field, and the resulting representations are called Specht modules. The Young symmetrizer is named after British mathematician Alfred Young.

In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory.

<span class="mw-page-title-main">Dihedral group of order 6</span> Non-commutative group with 6 elements

In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group.

In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century. The algebraic ring structure is a more recent development, due to Solomon (1967).

In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.

In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings, it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal.

In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group. They were introduced in where they were used to describe a class of finite p-groups whose structure was sufficiently similar to that of finite abelian p-groups, the so-called, regular p-groups. The relationship between power and commutator structure forms a central theme in the modern study of p-groups, as exemplified in the work on uniformly powerful p-groups.

<span class="mw-page-title-main">Hyperoctahedral group</span> Group of symmetries of an n-dimensional hypercube

In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parameter n, the dimension of the hypercube.

In mathematics, the O'Nan–Scott theorem is one of the most influential theorems of permutation group theory; the classification of finite simple groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric group. It appeared as an appendix to a paper by Leonard Scott written for The Santa Cruz Conference on Finite Groups in 1979, with a footnote that Michael O'Nan had independently proved the same result. Michael Aschbacher and Scott later gave a corrected version of the statement of the theorem.

<span class="mw-page-title-main">Affine symmetric group</span> Mathematical structure

The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects. In addition to this geometric description, the affine symmetric groups may be defined in other ways: as collections of permutations (rearrangements) of the integers that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied in combinatorics and representation theory.

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)