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Statement

Let H and A be groups, let K = AH 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 ⋅ ϕ ) ( g ) = ϕ ( h − 1 g ) , {\displaystyle (h\cdot \phi )(g)=\phi (h^{-1}g),} where ϕ ∈ K , {\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 A ≀ H . {\displaystyle A\wr H.} The group K = AH (which is isomorphic to { ( ϕ , 1 ) ∈ A ≀ H : ϕ ∈ K } {\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 θ : G → A ≀ H {\displaystyle \theta :G\to A\wr H} such that A maps surjectively onto im ( θ ) ∩ K . {\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 ψ : G → H {\displaystyle \psi :G\to H} whose kernel is A. Choose a set T = { t u : u ∈ H } {\displaystyle T=\{t_{u}:u\in H\}} of (right) coset representatives of A in G, where ψ ( t u ) = u . {\displaystyle \psi (t_{u})=u.} Then for all x in G, t u − 1 x t ψ ( x ) − 1 u ∈ ker ⁡ ψ = A . {\displaystyle t_{u}^{-1}xt_{\psi (x)^{-1}u}\in \ker \psi =A.} For each x in G, we define a function f x : H → A {\displaystyle f_{x}:H\to A} such that f x ( u ) = t u − 1 x t ψ ( x ) − 1 u . {\displaystyle f_{x}(u)=t_{u}^{-1}xt_{\psi (x)^{-1}u}.} Then the embedding θ {\displaystyle \theta } is given by θ ( x ) = ( f x , ψ ( x ) ) ∈ A ≀ H . {\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 θ ( x ) θ ( y ) = ( f x ( ψ ( x ) . f y ) , ψ ( x y ) ) . {\displaystyle \theta (x)\theta (y)=(f_{x}(\psi (x).f_{y}),\psi (xy)).} Now ψ ( x ) . f y ( u ) = f y ( ψ ( x ) − 1 u ) , {\displaystyle \psi (x).f_{y}(u)=f_{y}(\psi (x)^{-1}u),} so for all u in H,

f x ( u ) ( ψ ( x ) . f y ( u ) ) = t u − 1 x t ψ ( x ) − 1 u t ψ ( x ) − 1 u − 1 y t ψ ( y ) − 1 ψ ( x ) − 1 u = t u x y t ψ ( x y ) − 1 u − 1 , {\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 fx fy = fxy. Hence θ {\displaystyle \theta } is a homomorphism as required.

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

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.

References

1. Kaloujnine & Krasner (1951a). - 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= ↩

2. Dixon & Mortimer (1996, p. 47). - Dixon, John; Mortimer, Brian (1996). Permutation Groups. Springer. ISBN 978-0387945996. https://archive.org/details/permutationgroup0000dixo ↩

3. Dixon & Mortimer (1996, pp. 47–48). - Dixon, John; Mortimer, Brian (1996). Permutation Groups. Springer. ISBN 978-0387945996. https://archive.org/details/permutationgroup0000dixo ↩

4. Kaloujnine & Krasner (1951b). - 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 ↩