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 multiplication. This action extends naturally to an action of H on K, defined as $(h\cdot \phi )(g)=\phi (h^{-1}g),$ where $\phi \in K,$ and g and h are both in H. This is an automorphism of K, so we can construct the semidirect product K ⋊ H, which is termed the regular wreath product, and denoted A Wr H or $A\wr H.$ The group K = A^H (which is isomorphic to $\{(\phi ,1)\in A\wr H:\phi \in K\}$) 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, $t_{u}^{-1}xt_{\psi (x)^{-1}u}\in \ker \psi =A.$ For each x in G, we define a function $f_{x}:H\to A$ such that $f_{x}(u)=t_{u}^{-1}xt_{\psi (x)^{-1}u}.$ 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 $\theta (x)\theta (y)=(f_{x}(\psi (x).f_{y}),\psi (xy)).$ Now $\psi (x).f_{y}(u)=f_{y}(\psi (x)^{-1}u),$ so for all u in H,

$f_{x}(u)(\psi (x).f_{y}(u))=t_{u}^{-1}xt_{\psi (x)^{-1}u}t_{\psi (x)^{-1}u}^{-1}yt_{\psi (y)^{-1}\psi (x)^{-1}u}=t_{u}xyt_{\psi (xy)^{-1}u}^{-1},$

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 $t_{u}^{-1}xt_{\psi (x)^{-1}u}=t_{u}^{-1}yt_{\psi (y)^{-1}u},$ but we can cancel $t_{u}^{-1}$ and $t_{\psi (x)^{-1}u}=t_{\psi (y)^{-1}u}$ 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)).

Bibliography

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