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 [math]\displaystyle{ \phi(g).h=\phi(gh^{-1}), }[/math] where [math]\displaystyle{ \phi\in K, }[/math] 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 [math]\displaystyle{ A\wr H. }[/math] The group K = A^H (which is isomorphic to [math]\displaystyle{ \{(f_x,1)\in A\wr H:x\in K\} }[/math]) 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 [math]\displaystyle{ \theta:G\to A\wr H }[/math] such that A maps surjectively onto [math]\displaystyle{ \text{im}(\theta)\cap K. }[/math] 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 [math]\displaystyle{ \psi:G\to H }[/math] whose kernel is A. Choose a set [math]\displaystyle{ T=\{t_u:u\in H\} }[/math] of (right) coset representatives of A in G, where [math]\displaystyle{ \psi(t_u)=u. }[/math] Then for all x in G, [math]\displaystyle{ t_u x t^{-1}_{u\psi(x)}\in\ker \psi=A. }[/math] For each x in G, we define a function f_x: H → A such that [math]\displaystyle{ f_x(u)=t_u x t^{-1}_{u\psi(x)}. }[/math] Then the embedding [math]\displaystyle{ \theta }[/math] is given by [math]\displaystyle{ \theta(x)=(f_x,\psi(x))\in A\wr H. }[/math]

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

[math]\displaystyle{ 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)}, }[/math]

so f_x f_y = f_xy. Hence [math]\displaystyle{ \theta }[/math] is a homomorphism as required.

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

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. https://archive.org/details/permutationgroup0000dixo. 
• 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. http://pub.acta.hu/acta/showCustomerArticle.action?id=6063&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=23f6740691db44c1&style=. 
• 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. http://pub.acta.hu/acta/showCustomerArticle.action?id=6072&dataObjectType=article. 
• Praeger, Cheryl; Schneider, Csaba (2018). Permutation groups and Cartesian Decompositions. Cambridge University Press. ISBN 978-0521675062. https://books.google.com/books?id=ISZaDwAAQBAJ.