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.[1] 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 ${\displaystyle \phi (g).h=\phi (gh^{-1}),}$ where ${\displaystyle \phi \in K,}$ 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 ${\displaystyle A\wr H.}$ The group K = A^H (which is isomorphic to ${\displaystyle \{(f_{x},1)\in A\wr H:x\in K\}}$) 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 ${\displaystyle \theta :G\to A\wr H}$ such that A maps surjectively onto ${\displaystyle {\text{im}}(\theta )\cap K.}$[2] 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.[3]

Define a homomorphism ${\displaystyle \psi :G\to H}$ whose kernel is A. Choose a set ${\displaystyle T=\{t_{u}:u\in H\}}$ of (right) coset representatives of A in G, where ${\displaystyle \psi (t_{u})=u.}$ Then for all x in G, ${\displaystyle t_{u}xt_{u\psi (x)}^{-1}\in \ker \psi =A.}$ For each x in G, we define a function f_x: H → A such that ${\displaystyle f_{x}(u)=t_{u}xt_{u\psi (x)}^{-1}.}$ Then the embedding ${\displaystyle \theta }$ is given by ${\displaystyle \theta (x)=(f_{x},\psi (x))\in A\wr H.}$

We now prove that this is a homomorphism. If x and y are in G, then ${\displaystyle \theta (x)\theta (y)=(f_{x}(f_{y}.\psi (x)^{-1}),\psi (xy)).}$ Now ${\displaystyle f_{y}(u).\psi (x)^{-1}=f_{y}(u\psi (x)),}$ so for all u in H,

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

so f_x f_y = f_xy. Hence ${\displaystyle \theta }$ is a homomorphism as required.

The homomorphism is injective. If ${\displaystyle \theta (x)=\theta (y),}$ then both f_x(u) = f_y(u) (for all u) and ${\displaystyle \psi (x)=\psi (y).}$ Then ${\displaystyle t_{u}xt_{u\psi (x)}^{-1}=t_{u}yt_{u\psi (y)}^{-1},}$ but we can cancel t_u and ${\displaystyle t_{u\psi (x)}^{-1}=t_{u\psi (y)}^{-1}}$ from both sides, so x = y, hence ${\displaystyle \theta }$ is injective. Finally, ${\displaystyle \theta (x)\in K}$ precisely when ${\displaystyle \psi (x)=1,}$ in other words when ${\displaystyle x\in A}$ (as ${\displaystyle A=\ker \psi }$).

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).[4] In this case, G is isomorphic to a subgroup of the regular wreath product A Wr (G/Core(A)).

References

1. Kaloujnine & Krasner (1951a).
2. Dixon & Mortimer (1996, p. 47).
3. Dixon & Mortimer (1996, pp. 47–48).
4. Kaloujnine & Krasner (1951b).

Bibliography

• Dixon, John; Mortimer, Brian (1996). Permutation Groups. Springer. ISBN 978-0387945996.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)
• Praeger, Cheryl; Schneider, Csaba (2018). Permutation groups and Cartesian Decompositions. Cambridge University Press. ISBN 978-0521675062.