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Let $G$ be a finite group.

It is well-known that if $G$ is an $M$-group, then $G$ is solvable, due to Taketa. The converse is not necessarily true: $SL(2,3)$ is solvable, but not an $M$-group.However, Dade proved that every solvable group embeds into an $M$-group.

The proof uses induction and the wreath product to show that for any solvable group $G$ there exists an $M$-group $G'$ such that $G$ is a subgroup of $G'$.

I think that results about the structure of $G'$ are probably too difficult, so I am curious about the index of $G$ in $G'$. The use of the wreath product in the existence proof suggests that $G'$ could be very large. Is someone aware of results in this direction? Or perhaps a more constructive proof than Dade's?

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Isn't the proof constructive, in that it gives you an iterated wreath product of prime-power groups? – Steve D Apr 30 '10 at 15:07
Lots of existence proofs use wreath products to find some large containing group, but they are very rarely even the right order of magnitude for a "minimal" embedding. In this case, S3 gets embedded in 3 wr 2 of order 18, and S4 gets embedded in (2 wr 2 wr 3 wr 2) of order 4718592. Even using the wreath product to extend the embedding, one could probably do quite a bit better than this. Of course if he is just asking for the index from Dade's construction, then yes it is not too hard to bound. – Jack Schmidt Apr 30 '10 at 15:45
Yes, sorry I should have been more clear. I am looking for 'better' -that is, smaller order- embedding results. – Johannes Wachs Apr 30 '10 at 18:08
While it is well-known that an $M$-group is solvable, the definition of $M$-group is not well-known. Somebody care to explain, please? – Guntram May 1 '10 at 13:47
The $M$ stands for monomial: Isaac's Character Theory of Finite Groups is an excellent reference. – Johannes Wachs May 1 '10 at 20:53

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