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Here is the statement on my exercise sheet :

Let $G$ be a finite group, $p$ the smallest prime factor of $\vert G \vert$ and $H$ a subgroup of $G$ whose index is $p$. Then $H$ is a normal subgroup of $G$.

I already know how to prove this powerful statement. But I searched on the literature to find more about that nomination but I could not find anything relevant. It often brought me back to some Frobenius results but nothing more...

If someone has references about it or has already heard about the name of that lemma, it would be great to share !

Thanks in advance !

NB : enter image description here

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    $\begingroup$ Who told you it was called Ore's lemma? The only Ore I know was publishing around the middle of the 20th century. This result is decades older than that. $\endgroup$ Oct 25 at 20:37
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    $\begingroup$ Sorry, I misread your comment as 'I was the lecturer'. $\endgroup$ Oct 25 at 20:43
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    $\begingroup$ This appears to be a CNRS thing. I'd be astonished if it were by Ore. The only Ore I know is Øystein Ore, who was born in 1899. It's always difficult to find stuff in Burnside's book, so I haven't found it in there. Rose doesn't attribute it to anyone. I cannot find it in Huppert either, unfortunately. I'm actually having trouble finding it in older textbooks like Hall and Scott. I still don't believe it's due to Ore, but I cannot find much about it. $\endgroup$ Oct 25 at 21:04
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    $\begingroup$ I found it only under the name Lemme d'Ore as @Maman said: ref. Francinou, Gianella, Exercices de mathématiques pour l'agrégation : Algèbre 1. and math.ens.fr/~debarre/TDB2.pdf $\endgroup$ Oct 25 at 21:12
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    $\begingroup$ There is a thread on this issue at les-mathematiques.net/phorum/read.php?3,329558,329678 see in particular the Jan 29 2007 post. This post suggests that Ore may have proved a generalization. Group theory texts by Hall and Debrueil are cited as potential sources of the attribution (I do not have these books), as is the Francinou problem book. $\endgroup$ Oct 25 at 21:13
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I have found the original article of Ore. On p439, top, of Contributions to the Theory of Groups of Finite Order by Ore found here, he develops a theory of quasi-normal subgroups, and from that deduces the result we are talking about, due to Frobenius. Frobenius's result is from 1895, Ore's paper is from 1939.

It is Frobenius's lemma. To name it after Ore is inaccurate. That's like naming Kempe--Heawood's five-colour theorem after Appel and Haken because they proved a better result.

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  • $\begingroup$ Thank you for your research ! It will be helpful for my culture ! No surprised that this result is linked with Frobenius work ! $\endgroup$
    – Maman
    Oct 25 at 21:44
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    $\begingroup$ The reference one finds in Ore's paper is "G. Frobenius, Über endliche Gruppen, Sitzungsber. Akad. Berlin, 1895 (I), pp. 163-194." Frobenius' lemma is III on p. 171, as follows: Let $p$ be the smallest prime divisor of $m$, and let $d \leq p$. Then in any finite group of order $md$, a subgroup of order $m$ is normal. (So slightly more general.) $\endgroup$ Oct 26 at 12:05

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