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 by a group is isomorphic to a subgroup of the regular wreath product The theorem is named for the fact that the group is said to be universal with respect to all extensions of by

Statement

Let and be groups, let be the set of all functions from to and consider the action of on itself by right multiplication. This action extends naturally to an action of on defined by where and and are both in This is an automorphism of so we can define the semidirect product called the regular wreath product, and denoted or The group is called the base group of the wreath product.
The Krasner–Kaloujnine universal embedding theorem states that if has a normal subgroup and then there is an injective homomorphism of groups such that maps surjectively onto This is equivalent to the wreath product having a subgroup isomorphic to where is any extension of by

Proof

This proof comes from Dixon–Mortimer.
Define a homomorphism whose kernel is Choose a set of coset representatives of in where Then for all in For each in we define a function such that Then the embedding is given by
We now prove that this is a homomorphism. If and are in then Now so for all in
so Hence is a homomorphism as required.
The homomorphism is injective. If then both and Then but we can cancel and from both sides, so hence is injective. Finally, precisely when in other words when .

Generalizations and related results