This question is inspired by Terry Tao's [blog post](http://terrytao.wordpress.com/2014/10/06/a-trivial-generalisation-of-cayleys-theorem/) and the comments there. I have always cited M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de groupes III", Acta Sci. Math. Szeged 14, pp. 69-82 (1951) as the source of the result that if $G$ is a group and $H$ is a subgroup, then the regular action of $G$ on itself is inside of the action of the wreath product $H\wr (G/\mathrm{Core}(H))$ on $H\times G/H$ via the identification of $G$ with $H\times G/H$ using a transversal $T$ for $G/H$ and that replacing $T$ by a different transversal changes the embedding of $G$ by an inner automorphism of the wreath product. 


I am aware that this result is implicit in Frobenius's theory of induced representations and the theory of monomial representations of groups (e.g. Chapter 4 this paper of [Ore](http://www.ams.org/journals/tran/1942-051-00/S0002-9947-1942-0005739-6/S0002-9947-1942-0005739-6.pdf) although $H$ is assumed finite index there) but these authors did not use the wreath product language to the best of my knowledge. 


Many people refer only to the special case where $N$ is normal as the Krasner-Kaloujnine embedding. I am now wondering which version they actually stated. The original paper does not seem easy to access by the obvious Google search. 



What version of the wreath product embedding theorem is actually stated in the famous paper of Kaloujnine and Krasner?
The original paper (part II of a three part series) is available [here](http://acta.fyx.hu/acta/showCustomerArticle.action?id=6063&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=30c6b17da837ee4c&style=). Page 47 lists the theorem and discusses in a footnote the relation with Ore's earlier work.


From what I understand, any series of subgroups is allowed in the embedding, they need not be normal.


A more limited version of their theorem is published in part III, available [here](http://acta.fyx.hu/acta/showCustomerArticle.action?id=6072&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=4919364f62e409ab&style=), and this has caused some [confusion](http://www.google.nl/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0CCoQFjAB&url=http%3A%2F%2Fjournals.cambridge.org%2Farticle_S144678870001483X&ei=XQxFVJq0C8PqOKKLgfgI&usg=AFQjCNHQqwkXO_zu6u385i-N1f_i0GJ4Tg&sig2=PpWVQCSqmEyOC0nSHZ9WPg).


