Universal embedding theorem explained

The universal embedding theorem, or Krasner–Kaloujnine universal embedding theorem, is a theorem from the mathematical discipline of group theory first published in 1951 by Marc Krasner and Lev Kaluznin.^[1] The theorem states that any group extension of a group by a group is isomorphic to a subgroup of the regular wreath product The theorem is named for the fact that the group is said to be universal with respect to all extensions of by

Statement

Let and be groups, let be the set of all functions from to and consider the action of on itself by right multiplication. This action extends naturally to an action of on defined by

\phi(g).h=\phi(gh^-1),

where

\phi\inK,

and and are both in This is an automorphism of so we can define the semidirect product called the regular wreath product, and denoted or

A\wrH.

The group (which is isomorphic to

\{(f_x,1)\inA\wrH:x\inK\}

) is called the base group of the wreath product.

The Krasner–Kaloujnine universal embedding theorem states that if has a normal subgroup and then there is an injective homomorphism of groups

\theta:G\toA\wrH

such that maps surjectively onto

im(\theta)\capK.

^[2] This is equivalent to the wreath product having a subgroup isomorphic to where is any extension of by

Proof

This proof comes from Dixon–Mortimer.^[3]

Define a homomorphism

\psi:G\toH

whose kernel is Choose a set

T=\{t_u:u\inH\}

of (right) coset representatives of in where

\psi(t_u)=u.

Then for all in

t_ux

	-1
t
	u\psi(x)

\in\ker\psi=A.

For each in we define a function such that

f_x(u)=t_ux

	-1
t
	u\psi(x)

Then the embedding

\theta

is given by

\theta(x)=(f_{x,\psi(x))\in}A\wrH.

We now prove that this is a homomorphism. If and are in then

\theta(x)\theta(y)=(f_x(f

	-1

	y.\psi(x)

),\psi(xy)).

Now

	-1
f
	y(u).\psi(x)

=f_y(u\psi(x)),

so for all in

f_x(u)(f_{y(u).\psi(x))}=t_ux

	-1
t
	u\psi(x)

t_u\psi(x)y

	-1
t
	u\psi(x)\psi(y)

=t_uxy

	-1
t
	u\psi(xy)

so Hence

\theta

is a homomorphism as required.

The homomorphism is injective. If

\theta(x)=\theta(y),

then both (for all u) and

\psi(x)=\psi(y).

Then

t_ux

	-1
t
	u\psi(x)

=t_uy

	-1
t
	u\psi(y)

but we can cancel and

	-1
t
	u\psi(x)

	-1
=t
	u\psi(y)

from both sides, so hence

\theta

is injective. Finally,

\theta(x)\inK

precisely when

\psi(x)=1,

in other words when

x\inA

(as

A=\ker\psi

Generalizations and related results

The Krohn–Rhodes theorem is a statement similar to the universal embedding theorem, but for semigroups. A semigroup is a divisor of a semigroup if it is the image of a subsemigroup of under a homomorphism. The theorem states that every finite semigroup is a divisor of a finite alternating wreath product of finite simple groups (each of which is a divisor of) and finite aperiodic semigroups.
An alternate version of the theorem exists which requires only a group and a subgroup (not necessarily normal).^[4] In this case, is isomorphic to a subgroup of the regular wreath product

Bibliography

Book: Dixon . John . Mortimer . Brian . Permutation Groups . registration . 1996 . Springer . 978-0387945996 .
Kaloujnine . Lev . Krasner . Marc . Produit complet des groupes de permutations et le problème d'extension de groupes II . Acta Sci. Math. Szeged . 1951a . 14 . 39–66.
Kaloujnine . Lev . Krasner . Marc . Produit complet des groupes de permutations et le problème d'extension de groupes III . Acta Sci. Math. Szeged . 1951b . 14 . 69–82 .
Book: Praeger . Cheryl . Schneider . Csaba . Permutation groups and Cartesian Decompositions . 2018 . Cambridge University Press . 978-0521675062 . PS.

Universal embedding theorem explained

Statement

Proof

Generalizations and related results

Bibliography

Notes and References