A note on the Kaloujnine-Krasner theorem

Abstract

The celebrated Kaloujnine-Krasner theorem associates, with a short exact sequence of groups and a section , an embedding of G into the (unrestricted) wreath product of N and H. Given two groups H and N, a short exact sequence as above is called an extension of H by N, denoted by . Moreover, one says that two extensions and of H by N are equivalent if there exists a group isomorphism such that and . We say that two embeddings and are equivalent if there exists a group isomorphism such that . We show that two extensions and are equivalent if and only if the embeddings and , associated with any two sections and via the Kaloujnine-Krasner theorem, are equivalent.