Galois’ last letter

By lieven

Trinities

Galois’ last letter
Arnold’s trinities
Arnold’s trinities version 2.0
the buckyball symmetries
Klein’s dessins d’enfant and the buckyball
the buckyball curve

“Ne pleure pas, Alfred ! J’ai besoin de tout mon courage pour mourir à vingt ans!”

We all remember the last words of Evariste Galois to his brother Alfred. Lesser known are the mathematical results contained in his last letter, written to his friend Auguste Chevalier, on the eve of his fatal duel. Here the final sentences :

Tu prieras publiquement Jacobi ou Gauss de donner leur avis non sur la verite, mais sur l’importance des theoremes.
Apres cela il se trouvera, j’espere, des gens qui trouvent leur profis a dechiffrer tout ce gachis.
Je t’embrasse avec effusion. E. Galois, le 29 Mai 1832

A major result contained in this letter concerns the groups $L_2(p)=PSL_2(\mathbb{F}_p)$ , that is the group of $2 \times 2$ matrices with determinant equal to one over the finite field $\mathbb{F}_p$ modulo its center. L_2(p) is known to be simple whenever $p \geq 5$ . Galois writes that L_2(p) cannot have a non-trivial permutation representation on fewer than p+1 symbols whenever p > 11 and indicates the transitive permutation representation on exactly symbols in the three ‘exceptional’ cases p=5,7,11 .

Let $\alpha = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and consider for p=5,7,11 the involutions on $\mathbb{P}^1_{\mathbb{F}_p} = \mathbb{F}_p \cup \{ \infty \}$ (on which L_2(p) acts via Moebius transformations)

$\pi_5 = (0,\infty)(1,4)(2,3) \quad \pi_7=(0,\infty)(1,3)(2,6)(4,5) \quad \pi_{11}=(0,\infty)(1,6)(3,7)(9,10)(5,8)(4,2)$

(in fact, Galois uses the involution $~(0,\infty)(1,2)(3,6)(4,8)(5,10)(9,7)$ for p=11 ), then L_2(p) leaves invariant the set consisting of the involutions $\Pi = \{ \alpha^{-i} \pi_p \alpha^i~:~1 \leq i \leq p \}$ . After mentioning these involutions Galois merely writes :

Ainsi pour le cas de , l’equation modulaire s’abaisse au degre p.
En toute rigueur, cette reduction n’est pas possible dans les cas plus eleves.

Alternatively, one can deduce these permutation representation representations from group isomorphisms. As $L_2(5) \simeq A_5$ , the alternating group on 5 symbols, L_2(5) clearly acts transitively on 5 symbols.

Similarly, for p=7 we have $L_2(7) \simeq L_3(2)$ and so the group acts as automorphisms on the projective plane over the field on two elements $\mathbb{P}^2_{\mathbb{F}_2}$ aka the Fano plane, as depicted on the left.

This finite projective plane has 7 points and 7 lines and L_3(2) acts transitively on them.

For p=11 the geometrical object is a bit more involved. The set of non-squares in $\mathbb{F}_{11}$ is

$\{ 1,3,4,5,9 \}$

and if we translate this set using the additive structure in $\mathbb{F}_{11}$ one obtains the following 11 five-element sets

$\{ 1,3,4,5,9 \}, \{ 2,4,5,6,10 \}, \{ 3,5,6,7,11 \}, \{ 1,4,6,7,8 \}, \{ 2,5,7,8,9 \}, \{ 3,6,8,9,10 \},$

$\{ 4,7,9,10,11 \}, \{ 1,5,8,10,11 \}, \{ 1,2,6,9,11 \}, \{ 1,2,3,7,10 \}, \{ 2,3,4,8,11 \}$

and if we regard these sets as ‘lines’ we see that two distinct lines intersect in exactly 2 points and that any two distinct points lie on exactly two ‘lines’. That is, intersection sets up a bijection between the 55-element set of all pairs of distinct points and the 55-element set of all pairs of distinct ‘lines’. This is called the biplane geometry.

The subgroup of $S_{11}$ (acting on the eleven elements of $\mathbb{F}_{11}$ ) stabilizing this set of 11 5-element sets is precisely the group L_2(11) giving the permutation representation on 11 objects.

An alternative statement of Galois’ result is that for p > 11 there is no subgroup of L_2(p) complementary to the cyclic subgroup

$C_p = \{ \begin{bmatrix} 1 & x \\ 0 & 1 \end{bmatrix}~:~x \in \mathbb{F}_p \}$

That is, there is no subgroup such that set-theoretically $L_2(p) = F \times C_p$ (note this is of courese not a group-product, all it says is that any element can be written as g=f.c with $f \in F, c \in C_p$ .

However, in the three exceptional cases we do have complementary subgroups. In fact, set-theoretically we have

$L_2(5) = A_4 \times C_5 \qquad L_2(7) = S_4 \times C_7 \qquad L_2(11) = A_5 \times C_{11}$

and it is a truly amazing fact that the three groups appearing are precisely the three Platonic groups!

Recall that here are 5 Platonic (or Scottish) solids coming in three sorts when it comes to rotation-automorphism groups : the tetrahedron (group A_4 ), the cube and octahedron (group S_4 ) and the dodecahedron and icosahedron (group A_5 ). The “4″ in the cube are the four body diagonals and the “5″ in the dodecahedron are the five inscribed cubes.

That is, our three ‘exceptional’ Galois-groups correspond to the three Platonic groups, which in turn correspond to the three exceptional Lie algebras E_6,E_7,E_8 via McKay correspondence (wrt. their 2-fold covers). Maybe I’ll detail this latter connection another time. It sure seems that surprises often come in triples…

Finally, it is well known that $L_2(5) \simeq A_5$ is the automorphism group of the icosahedron (or dodecahedron) and that L_2(7) is the automorphism group of the Klein quartic.

So, one might ask : is there also a nice curve connected with the third group L_2(11) ? Rumour has it that this is indeed the case and that the curve in question has genus 70… (to be continued).

Reference

Bertram Kostant, “The graph of the truncated icosahedron and the last letter of Galois”

Next in series

Galois, geometry, groups, Klein, permutation representation, representations

9 comments
Posted in groups
Written on Thu, 12 June 2008 at 7:53 pm
Tags: Galois, geometry, groups, Klein, permutation representation, representations
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9 Responses to “Galois’ last letter”

Kea Says:
June 13th, 2008 at 12:39 am
Genus 70 … Moonshine, Moonshine! Cool! It sure seems that surprises often come in triples… Yes, just like neutrino masses! And ternary logic!
Carl Brannen Says:
June 13th, 2008 at 10:54 am
If there is an after-life, and I am so lucky as to meet Galois, I am going to tell him that each time I was reminded of his early demise my thought was “what an idiot!” The reason the military drafts people that age is because when humans reach 25 they’re no longer as willing to die for causes, lost or won, that are quickly forgotten.
k. sris Says:
June 16th, 2008 at 5:37 pm
i enjoy reading your blog. can you point me to an easy reference ideally at undergrad level.

k
David Corfield Says:
June 16th, 2008 at 7:33 pm
Have you read Arnold on surprising triples here? Page 10 discusses yours.
lieven Says:
June 17th, 2008 at 7:04 am
@david : no, i havent read Arnold’s paper yet, thanks for the reference, ill check it out.

@k. sris : Platonic solids and groups are usually covered in a first year course in group theory and is included in many textbooks eg. ledermann. McKay correspondence is an application to the representation theory of finite groups (in antwerp i teach this the second year). perhaps ill post about it later. if you like to have a less dry introduction to finite groups (including details on L_2(p) and the Platonics) i can advise strongly “Adventures in group theory” by David Joyner (published by John Hopkins University).
k. sris Says:
June 17th, 2008 at 3:51 pm
thanks. i look forward to your post.

i got interested after reading the nice groups and their graphs by magnus and grossman. cayley table is so easy to construct with the graphs. i will check out adventures in groups theory.

i have another question that i have been wondering for a while. what is the motivation and history for considering the conjuagte a^{-1}ba. i kind of know that conjugate of b is b modified by a and that it in some kind of way measures the commutativity of a and b. but i like to know how does one come up with this particular combination?
David Corfield Says:
June 17th, 2008 at 9:57 pm
The comments seem to be closed on your Arnold post, so I’ll pose a question here. Arnold, as you note, sees his trinities as intimately connected - ‘functorial’, ‘commutative diagrams’, etc. So does your alteration in 1 to include the octonions and exclude the reals suggest you don’t find that plausible? Trinity 16 hasn’t been linked to other trinities, has it?
Kea Says:
June 18th, 2008 at 3:23 am
David or Lieven, is there any chance you could convert Arnold’s paper to pdf so that more of us could download it?
lieven Says:
June 18th, 2008 at 6:58 am
@Kea, sorry about that. The pdf-conversion can be found here.

@David, thanks for mentioning the problem with the comment of the Arnold post. Ill fix it and reply to your comment there, later today