A note on the Kaloujnine-Krasner theorem.

The celebrated Kaloujnine-Krasner theorem associates, with a short exact sequence 1 → N → ι G → π H → 1 of groups and a section s : H → G , an embedding Φ : G → N ≀ H 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 (G ; ι , π) . Moreover, one says that two extensions (G 1 ; ι 1 , π 1) and (G 2 ; ι 2 , π 2) of H by N are equivalent if there exists a group isomorphism η : G 1 → G 2 such that ι 2 = η ° ι 1 and π 1 = π 2 ° η . We say that two embeddings Φ 1 : G 1 → N ≀ H and Φ 2 : G 2 → N ≀ H are equivalent if there exists a group isomorphism η : G 1 → G 2 such that Φ 1 = Φ 2 ° η . We show that two extensions (G 1 ; ι 1 , π 1) and (G 2 ; ι 2 , π 2) are equivalent if and only if the embeddings Φ 1 and Φ 2 , associated with any two sections s 1 : H → G 1 and s 2 : H → G 2 via the Kaloujnine-Krasner theorem, are equivalent. [ABSTRACT FROM AUTHOR]