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This question is inspired by Terry Tao's blog post 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 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.

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The original paper (part II of a three part series) is available here. 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, and this has caused some confusion.

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  • $\begingroup$ Thanks for locating it. I will have to read this on a bigger screen then my phone. $\endgroup$ Oct 20, 2014 at 13:27
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    $\begingroup$ I read the paper about 20 years ago for my thesis. The way I remember their result (and what I definitively used) is that the subgroups don't need to be normal. I'm still puzzled why the result (which is induction for permutation representations) is not put in (more) textbooks. $\endgroup$
    – ahulpke
    Oct 20, 2014 at 15:12

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