Skip to content →

Galois’ last letter

“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 L2(p)=PSL2(Fp), that is the group of 2×2 matrices with determinant equal to one over the finite field Fp modulo its center. L2(p) is known to be simple whenever p5. Galois writes that L2(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 p symbols in the three ‘exceptional’ cases p=5,7,11.

Let α=[1101] and consider for p=5,7,11 the involutions on PFp1=Fp (on which L2(p) acts via Moebius transformations)

π5=(0,)(1,4)(2,3)π7=(0,)(1,3)(2,6)(4,5)π11=(0,)(1,6)(3,7)(9,10)(5,8)(4,2)

(in fact, Galois uses the involution  (0,)(1,2)(3,6)(4,8)(5,10)(9,7) for p=11), then L2(p) leaves invariant the set consisting of the p involutions Π=αiπpαi : 1ip. After mentioning these involutions Galois merely writes :

Ainsi pour le cas de p=5,7,11, 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 L2(5)A5, the alternating group on 5 symbols, L2(5) clearly acts transitively on 5 symbols.

Similarly, for p=7 we have L2(7)L3(2) and so the group acts as automorphisms on the projective plane over the field on two elements PF22 aka the Fano plane, as depicted on the left.

This finite projective plane has 7 points and 7 lines and L3(2) acts transitively on them.

For p=11 the geometrical object is a bit more involved. The set of non-squares in F11 is

1,3,4,5,9

and if we translate this set using the additive structure in F11 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 S11 (acting on the eleven elements of F11) stabilizing this set of 11 5-element sets is precisely the group L2(11) giving the permutation representation on 11 objects.

An alternative statement of Galois’ result is that for p>11 there is no subgroup of L2(p) complementary to the cyclic subgroup

Cp=[1x01] : xFp

That is, there is no subgroup such that set-theoretically L2(p)=F×Cp (note this is of courese not a group-product, all it says is that any element can be written as g=f.c with fF,cCp.

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

L2(5)=A4×C5L2(7)=S4×C7L2(11)=A5×C11

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 A4), the cube and octahedron (group S4) and the dodecahedron and icosahedron (group A5). 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 E6,E7,E8 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 L2(5)A5 is the automorphism group of the icosahedron (or dodecahedron) and that L2(7) is the automorphism group of the Klein quartic.

So, one might ask : is there also a nice curve connected with the third group L2(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”

Published in groups

Comments

Leave a Reply

Your email address will not be published. Required fields are marked *