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 ${\displaystyle (h\cdot \phi )(g)=\phi (h^{-1}g),}$ where ${\displaystyle \phi \in K,}$ 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 ${\displaystyle A\wr H.}$ The group K = A^H (which is isomorphic to ${\displaystyle \{(\phi ,1)\in A\wr H:\phi \in K\}}$) 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 ${\displaystyle \theta :G\to A\wr H}$ such that A maps surjectively onto ${\displaystyle {\text{im}}(\theta )\cap K.}$ 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 ${\displaystyle \psi :G\to H}$ 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, ${\displaystyle t_{u}^{-1}xt_{\psi (x)^{-1}u}\in \ker \psi =A.}$ For each x in G, we define a function ${\displaystyle f_{x}:H\to A}$ such that ${\displaystyle f_{x}(u)=t_{u}^{-1}xt_{\psi (x)^{-1}u}.}$ 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 ${\displaystyle \theta (x)\theta (y)=(f_{x}(\psi (x).f_{y}),\psi (xy)).}$ Now ${\displaystyle \psi (x).f_{y}(u)=f_{y}(\psi (x)^{-1}u),}$ so for all u in H,

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

so f_x f_y = f_xy. Hence ${\displaystyle \theta }$ 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 ${\displaystyle t_{u}^{-1}xt_{\psi (x)^{-1}u}=t_{u}^{-1}yt_{\psi (y)^{-1}u},}$ but we can cancel ${\displaystyle t_{u}^{-1}}$ and ${\displaystyle t_{\psi (x)^{-1}u}=t_{\psi (y)^{-1}u}}$ 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). In this case, G is isomorphic to a subgroup of the regular wreath product A Wr (G/Core(A)).

References

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. Archived from the original on 2024-11-19. Retrieved 2019-07-07.
• 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.