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I am reading Galois' last letter to Auguste Chevalier before the infamous duel.

https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100

What is the notation H' for a group H supposed to represent here? Later in the letter he uses the word "differentials". Surely these are not the same differentials from Calculus? And yet later, integrals are introduced. How do I make sense of these connections? How to make sense of what he means by "modular equation" and a sine function being related to permutations? What result of Legendre is he referencing on the 7th page for $\frac{\pi}{2}? I googled and found nothing of the sort.

Edit: I seem to have found the result referenced https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_algorithm it is Gauss-Legendre Derivative of the elliptic integral of the first kind

vallev
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  1. Regarding the $T', S'$ notation, Galois is talking about the decomposition of a group $G$ into left resp. right cosets with respect to a subgroup $H$. The notation $H, HS, HS', \dots$ is meant to indicate the list of cosets, so $S, S', \dots$ is meant to indicate some collection of elements of $G$ belonging to different cosets. It is a little confusing that he capitalizes them but with enough context it's basically clear what he means. So that stuff about proper decompositions is referring to when $H$ is normal.

  2. This is not my area, but I think the bit about modular equations is referring to the use of the $j$-invariant to construct polynomials with Galois group $PGL_2(\mathbb{Z}/n\mathbb{Z})$. I'm not sure what he means by "sine of the amplitude..." here but it could be referring to the elliptic sine.

  3. This is extremely not my area, but yes, he means differentials and integrals in the calculus sense. I believe this bit is referring to a circle of ideas involving hyperelliptic integrals, periods, and abelian varieties, but I'm not familiar enough with this stuff to say more.

Qiaochu Yuan
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