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.
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 right multiplication.
This action extends naturally to an action of H on K defined by
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
The group K = AH (which is isomorphic to
) 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
such that A maps surjectively onto
[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.
[3] Define a homomorphism
ψ :
Choose a set
of (right) coset representatives of A in G, where
ψ (
u ψ ( x )
∈ ker ψ =
For each x in G, we define a function fx: H → A such that
u ψ ( x )
, ψ ( x ) ) ∈
We now prove that this is a homomorphism.
ψ ( x
) , ψ ( x y ) ) .
ψ ( x
so for all u in H, so fx fy = fxy.
is a homomorphism as required.
The homomorphism is injective.
then both fx(u) = fy(u) (for all u) and
but we can cancel tu and
= ker ψ