Universal embedding theorem

Statement

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 $A \operatorname{Wr} H.$ The theorem is named for the fact that the group $A \operatorname{Wr} H$ is said to be universal with respect to all extensions of $H$ by $A.$