Universal embedding theorem

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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 = 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),} {\displaystyle (h\cdot \phi )(g)=\phi (h^{-1}g),} where ϕ ∈ K , {\displaystyle \phi \in 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.} {\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\}} {\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} {\displaystyle \theta :G\to A\wr H} such that A maps surjectively onto im ( θ ) ∩ K . {\displaystyle {\text{im}}(\theta )\cap 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} {\displaystyle \psi :G\to H} whose kernel is A. Choose a set T = { t u : u ∈ H } {\displaystyle T=\{t_{u}:u\in 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.} {\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.} {\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} {\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}.} {\displaystyle f_{x}(u)=t_{u}^{-1}xt_{\psi (x)^{-1}u}.} Then the embedding θ {\displaystyle \theta } {\displaystyle \theta } is given by θ ( x ) = ( f x , ψ ( x ) ) ∈ A ≀ H . {\displaystyle \theta (x)=(f_{x},\psi (x))\in A\wr 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)).} {\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),} {\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},} {\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 } {\displaystyle \theta } is a homomorphism as required.

The homomorphism is injective. If θ ( x ) = θ ( y ) , {\displaystyle \theta (x)=\theta (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).} {\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},} {\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}} {\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}} {\displaystyle t_{\psi (x)^{-1}u}=t_{\psi (y)^{-1}u}} from both sides, so x = y, hence θ {\displaystyle \theta } {\displaystyle \theta } is injective. Finally, θ ( x ) ∈ K {\displaystyle \theta (x)\in K} {\displaystyle \theta (x)\in K} precisely when ψ ( x ) = 1 , {\displaystyle \psi (x)=1,} {\displaystyle \psi (x)=1,} in other words when x ∈ A {\displaystyle x\in A} {\displaystyle x\in A} (as A = ker ⁡ ψ {\displaystyle A=\ker \psi } {\displaystyle A=\ker \psi }).

  • 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

Bibliography