Galois’ Last Letter (2008)

A major result contained in this letter concerns the groups $L 2(p)=PSL 2(mathbb p) $, that is the group of $2 times 2 $ matrices with determinant equal to one over the finite field $mathbb p $ modulo its center. 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. The subgroup of $S $ stabilizing this set of 11 5-element sets is precisely the group $L 2(11) $ giving the permutation representation on 11 objects. Finally, it is well known that $L 2(5) simeq A 5 $ is the automorphism group of the icosahedron and that $L 2(7) $ is the automorphism group of the Klein quartic. 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. Bertram Kostant, "The graph of the truncated icosahedron and the last letter of Galois".