Wreath product

≀ _X

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 are groups and J operates on a set Y, then an operation of J on (the group of all mappings from Y to G with pointwise connection) is induced by: ${\displaystyle G^{Y}}$

${\displaystyle \forall j\in J,f\in G^{Y}:(^{j}f)(y)=f(^{j^{-1}}y)}$

Each defines an automorphism of in this way . ${\displaystyle j\in J}$ ${\displaystyle G^{Y}}$

Thus, the wreath product can be defined as the semi-direct product of and J with respect to this very operation. Sometimes one also looks at the restricted wreath product. This is obtained by considering only the subgroup of the images, which disappear almost everywhere, instead of the group of all images from to . ${\displaystyle G\wr _{Y}J}$ ${\displaystyle G^{Y}}$ ${\displaystyle Y}$ ${\displaystyle G}$

properties

The cardinality of wreath products can immediately be derived from the definition: ${\displaystyle \left|G\wr _{Y}J\right|=\left|G\right|^{\left|Y\right|}\cdot \left|J\right|}$

Since each group operates on itself through left multiplication, it is often the case that only the corresponding wreath product is defined. It is also common to fix Y as a finite set and only allow subsets of Sym (n) with the canonical operation on Y for J. ${\displaystyle G\wr _{J}J}$ ${\displaystyle \{1,...,n\}}$

Operations

If G operates on a set X, this and the operation of J on Y induce an operation of on : ${\displaystyle G\wr _{Y}J}$ ${\displaystyle X\times Y}$

${\displaystyle \forall (x,y)\in X\times Y,(f,j)\in G\wr _{Y}J:^{(f,j)}(x,y):=(^{f(^{j}y)}x,^{j}y)}$

This operation is faithful / transitive if and only if the operations of G on X and J on Y are faithful / transitive.

Group expansions

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

So there is an exact sequence

${\displaystyle 1\longrightarrow N\longrightarrow ^{\!\!\!\!\!\!\!\!\!\iota }\ \,H\longrightarrow ^{\!\!\!\!\!\!\!\!\!\pi }\ \,Q\longrightarrow 1}$

In addition, a mapping is given that fulfills and assigns a fixed representative of its respective secondary class to each element. Must still apply . (If N is infinite, then such a function can possibly only be found with the axiom of choice ) ${\displaystyle q\colon H\to H}$ ${\displaystyle \forall g\in H:q(g)\iota (N)=g\iota (N)}$ ${\displaystyle \forall g\in H:q(g^{-1})=q(g)^{-1}}$

The embedding (Q operates on itself through left multiplication) is then given by: ${\displaystyle \phi \colon H\hookrightarrow N\wr _{Q}Q}$

${\displaystyle \phi (h):=(\sigma _{h},\pi (h))\,}$

It is defined as follows: ${\displaystyle \sigma _{h}\colon Q\to N}$

${\displaystyle \sigma _{h}(yN):=\iota ^{-1}(q(y^{-1})\cdot h\cdot q(h^{-1}y))}$

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

Examples

The p-Sylow groups of the symmetric group can be represented as iterated ring products of cyclic groups . ${\displaystyle S_{n}}$

To do this, a sequence of groups is defined recursively by and , where the operation from to is given by left multiplication. ${\displaystyle W_{p,0}:=\{1\}}$ ${\displaystyle W_{p,n+1}:=W_{p,n}\wr _{\mathbb {Z} _{p}}\mathbb {Z} _{p}}$ ${\displaystyle J=\mathbb {Z} _{p}}$ ${\displaystyle Y=\mathbb {Z} _{p}}$

If we represent n to the base p, i. H. as the sum with , then the p-Sylow groups of then are isomorphic to ${\displaystyle \sum _{i=0}^{k}{c_{i}p^{i}}}$ ${\displaystyle c_{i}\in \{0,...,p-1\}}$ ${\displaystyle S_{n}}$ ${\displaystyle \prod _{i=0}^{k}{W_{p,i}^{c_{i}}}}$

To the symbol

The vertical tilde that is used for the wreath product is in the Unicode block Mathematical Operators in position U + 2240, in TeX and LaTeX it can be represented with \wreathor \wr.

swell

1. ^ "Produit complet des groupes de permutations et probleme d'extension de groupes", L. Kaloujnine, M. Krasner - I, Acta Sci. Math. Szeged, 1950
2. ↑ Unicode Character 'WREATH PRODUCT' (U + 2240) , fileformat.info