Wreath product

The wreath product ( engl. wreath product ) is a term used in group theory and referred to a special semi- direct product of groups.

Definition

If G and J groups and operates J on a set Y, so by an operation of J on (the group of all mappings from Y to G with pointwise relation) induced by:

Each defined in this way an automorphism of.

Thus, the wreath product can be defined as the semi- product of direct and J with respect to this same operation. Sometimes you also considered the restricted wreath product. This is obtained by the group instead of all the pictures from looking after only the subset of images that vanish almost everywhere.

Properties

From the definition, the cardinality of wreath products can be immediately derived:

Since each group acts on itself by left multiplication, it is also often the case that only the corresponding wreath product is defined. Also common to Y set as a finite quantity and allow only subgroups of Sym ( n ) with the canonical operation on Y for J is.

Operations

G operates on a set X, then by and by the operation of J on Y induces an operation of at:

This operation is exactly true then / transitive if the operations of G on X and Y to Y are faithful / transitive.

Group extensions

If H is an extension of N by Q, then H can be as a subgroup of a wreath product of N and Q represent. This is perhaps one of the most important properties of wreath products, since every finite group can be represented by simple extensions of finite groups.

It may therefore be an exact sequence

Furthermore, an illustration is given, which satisfies each element and assigns a fixed representative of its respective coset. It must also apply. ( If N is infinite, so is a such a function may be found only with the axiom of choice )

The embedding (Q acts on itself by left multiplication ) is then given by:

Here is defined as follows:

This embedding goes back to L.Kaloujnine and M.Krasner.

Examples

The p- Sylowgruppen the symmetric group can be represented as iterated wreath products of cyclic groups.

To be defined recursively by a sequence of groups, and wherein the operation is given by means of links of at multiplication.

Is, n is the p base, i.e. as the sum of, the p- Sylowgruppen of then are isomorphic to

The symbol

The vertical tilde, which is used for the wreath product, is located in the Unicode block Mathematical operators to position U 2240, in TeX and LaTeX it can be represented with \ wreath or \ wr.

Swell

• Group Theory