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 $H$ by a group $A$ is isomorphic to a subgroup of the regular wreath product $A Wr H.$ The theorem is named for the fact that the group $A Wr H$ is said to be universal with respect to all extensions of $H$ by $A.$

Statement
Let $H$ and $A$ be groups, let $K = A^{H}$ be the set of all functions from $H$ to $A,$ and consider the action of $H$ on itself by right multiplication. This action extends naturally to an action of $H$ on $K$ defined by $$\phi(g).h=\phi(gh^{-1}),$$ where $$\phi\in K,$$ and $g$ and $h$ are both in $H.$ This is an automorphism of $K,$ so we can define the semidirect product $K ⋊ H$ called the regular wreath product, and denoted $A Wr H$ or $$A\wr H.$$ The group $K = A^{H}$ (which is isomorphic to $$\{(f_x,1)\in A\wr H:x\in K\}$$) is called the base group of the wreath product.

The Krasner–Kaloujnine universal embedding theorem states that if $G$ has a normal subgroup $A$ and $H = G/A,$ then there is an injective homomorphism of groups $$\theta:G\to A\wr H$$ such that $A$ maps surjectively onto $$\text{im}(\theta)\cap K.$$ This is equivalent to the wreath product $A Wr H$ having a subgroup isomorphic to $G,$ where $G$ is any extension of $H$ by $A.$

Proof
This proof comes from Dixon–Mortimer.

Define a homomorphism $$\psi:G\to H$$ whose kernel is $A.$ Choose a set $$T=\{t_u:u\in H\}$$ of (right) coset representatives of $A$ in $G,$ where $$\psi(t_u)=u.$$ Then for all $x$ in $G,$ $$t_u x t^{-1}_{u\psi(x)}\in\ker \psi=A.$$ For each $x$ in $G,$ we define a function $f_{x}: H → A$ such that $$f_x(u)=t_u x t^{-1}_{u\psi(x)}.$$ 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 $x$ and $y$ are in $G,$ then $$\theta(x)\theta(y)=(f_x(f_y.\psi(x)^{-1}),\psi(xy)).$$ Now $$f_y(u).\psi(x)^{-1}=f_y(u\psi(x)),$$ so for all $u$ in $H,$
 * $$f_x(u)(f_y(u).\psi(x)) = t_u x t^{-1}_{u\psi(x)} t_{u\psi(x)} y t^{-1}_{u\psi(x)\psi(y)}=t_u xy t^{-1}_{u\psi(xy)},$$

so $f_{x} f_{y} = f_{xy}.$ Hence $$\theta$$ is a homomorphism as required.

The homomorphism is injective. If $$\theta(x)=\theta(y),$$ then both $f_{x}(u) = f_{y}(u)$ (for all u) and $$\psi(x)=\psi(y).$$ Then $$t_u x t^{-1}_{u\psi(x)}=t_u y t^{-1}_{u\psi(y)},$$ but we can cancel $t_{u}$ and $$t^{-1}_{u\psi(x)}=t^{-1}_{u\psi(y)}$$ from both sides, so $x = y,$ 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 $S$ is a divisor of a semigroup $T$ if it is the image of a subsemigroup of $T$ under a homomorphism. The theorem states that every finite semigroup $S$ is a divisor of a finite alternating wreath product of finite simple groups (each of which is a divisor of $S$) and finite aperiodic semigroups.
 * An alternate version of the theorem exists which requires only a group $G$ and a subgroup $A$ (not necessarily normal). In this case, $G$ is isomorphic to a subgroup of the regular wreath product $A Wr (G/Core(A)).$