Universal embedding theorem

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. 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 multiplication. This action extends naturally to an action of on , defined as $(h\cdot \phi)(g)=\phi(h^g),$ where $\phi\in K,$ and and are both in This is an automorphism of so we can construct the semidirect product , which is termed the ''regular wreath product'', and denoted or $A\wr H.$ The group (which is isomorphic to $\$) 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 $\theta:G\to A\wr H$ such that maps surjectively onto $\text(\theta)\cap K.$ 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..

Define a homomorphism $\psi:G\to H$ whose kernel is Choose a set $T=\$ of (right) coset representatives of in where $\psi(t_u)=u.$ Then for all in $t_u^ x t_\in\ker \psi=A.$ For each in we define a function $f_x:H\to A$ such that $f_x(u)=t_u^ x t_.$ Then the embedding $\theta$ is given by $\theta(x)=(f_x,\psi(x))\in A\wr H.$

We now prove that this is a homomorphism. If and are in then $\theta(x)\theta(y)=(f_x(\psi(x).f_y),\psi(xy)).$ Now $\psi(x).f_y(u)=f_y(\psi(x)^u),$ so for all in
:$f_x(u)(\psi(x).f_y(u)) = t_u^ x t_ t_^ y t_=t_u xy t^_,$
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 $t_u^ x t_=t_u^ y t_,$ but we can cancel $t^_$ and $t_=t_$ from both sides, so hence $\theta$ is injective. Finally, $\theta(x)\in K$ precisely when $\psi(x)=1,$ in other words when $x\in A$ (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).. In this case, is isomorphic to a subgroup of the regular wreath product

References

Bibliography

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*{{cite book , last1=Praeger , first1=Cheryl , last2=Schneider , first2=Csaba , title=Permutation groups and Cartesian Decompositions , date=2018 , publisher=Cambridge University Press , isbn=978-0521675062 , url=https://books.google.com/books?id=ISZaDwAAQBAJ , ref=PS

Theorems in group theory