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 Template:Math by a group Template:Math is isomorphic to a subgroup of the regular wreath product Template:Math The theorem is named for the fact that the group Template:Math is said to be universal with respect to all extensions of Template:Math by Template:Math

Statement

Let Template:Math and Template:Math be groups, let Template:Math be the set of all functions from Template:Math to Template:Math and consider the action of Template:Math on itself by right multiplication. This action extends naturally to an action of Template:Math on Template:Math defined by $\varphi (g).h=\varphi (g{h}^{-1}),$ where $\varphi \in K,$ and Template:Math and Template:Math are both in Template:Math This is an automorphism of Template:Math so we can define the semidirect product Template:Math called the regular wreath product, and denoted Template:Math or $A\wr H.$ The group Template:Math (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 Template:Math has a normal subgroup Template:Math and Template:Math then there is an injective homomorphism of groups $\theta :G\to A\wr H$ such that Template:Math maps surjectively onto $\text{im}(\theta )\cap K.$ This is equivalent to the wreath product Template:Math having a subgroup isomorphic to Template:Math where Template:Math is any extension of Template:Math by Template:Math

Proof

This proof comes from Dixon–Mortimer.

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

$f_x(u)(f_y(u).\psi (x))=t_{u}x{t}_{u\psi (x)}^{-1}{t}_{u\psi (x)}y{t}_{u\psi (x)\psi (y)}^{-1}=t_{u}xy{t}_{u\psi (xy)}^{-1},$

so Template:Math Hence $\theta$ is a homomorphism as required.

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

Bibliography

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