Serre has been giving a series of lectures at Harvard for the last month, on finite groups in number theory. It started off with some ideas revolving around Chebotarev density, and recently moved into fusion (meaning conjugacy classes, not monoidal categories) and mod p representations. In between, he gave a neat self-contained talk about small finite groups, which really meant canonical structures on small finite sets.
He started by writing the numbers 2,3,4,5,6,7,8, indicating the sizes of the sets to be discussed, and then he tackled them in order.
There isn’t much to say about a set of size two. It forms a torsor for , and any action of a group on it yields a map .
If you haven’t seen the word torsor before, it is a fancy way of saying, “the underlying set (or space, etc.) of a group, with the multiplication action.” Baez has an introduction to them, and there is an article on Wikipedia, where it is also given names like heap and “groud”. You can find them in number theory in several guises. For example, if we are given a finite Galois field extension with Galois group , the field is a torsor for the group ring in -vector spaces (that is, a free module of rank one). Dually, the scheme is a torsor for the constant group scheme . I only had one successful research project as an undergrad, and it was a project with Lindsay Childs on classifying objects like these (in particular, fixing the field while varying the Hopf algebra), but I didn’t know that I was doing so at the time.
A set with three elements can be canonically attached to a (2,2) group as the set of nonidentity elements. More geometrically, one can take the hyperplane of sum zero elements, and there is a canonical map given by sending an element , where the zero is in the i-th coordinate. Permutations of this set naturally act by automorphisms on the group. .
At this point, Serre moved on to four element sets, probably due to some combination of limited time and the fact that that case is much richer. I thought I’d mention a couple other structures, using techniques he mentioned later. One can take rational points on the flag variety, i.e., with acting by linear fractional transformations, and the identification is the same as above, except instead of a nonzero point, we take the span of it. Alternatively, one can base change to , and take the six points in the complement of the coordinate planes. These are the nonzero points in the conic , and their coordinates are permutations of the nonzero elements of . We can send each element of to the pair of points whose i-th coordinate is one (or more canonically, fixed under Frobenius). acts on the six points by permuting the coordinates, and this action preserves the pairs. Switching to characteristic three, we can make the flag , and for , take (or switch “one” with “any nonzero element”). This is a torsor over , and we can identify an element of with the unique coordinate that is not equal to any other, since any representative of a point has two equal coordinates. Any identification induces an isomorphism between the symmetric group and the ax+b group . There is an analogous multiplicative structure over given by the “torus flag” . acts as a semidirect product of the multiplicative group with the inversion automorphism (which coincides with the Frobenius in this case). As a final example, the Fermat curve over the rationals has three rational points, namely those with zero in the i-th coordinate and plus or minus one in the others. It is a genus one curve, so it is a torsor for its Jacobian, an elliptic curve, and it has a canonical action of the semidirect product of its Jacobian with the minus one automorphism, whose group of rational points is isomorphic to .
A four element set has a structure of an affine space over a (2,2) group. As above, we take the hyperplane , but this time, there is a line inside: . The quotient is a (2,2) group
canonically attached to our set, and the three nontrivial elements correspond to the (2,2) partitions. We have a map , and is a torsor for , with a permutation action of by coordinate changes. If we view as an affine plane, then this action is by affine transformations, and yields an isomorphism . In ATLAS notation, this is and .
Switching to characteristic three, has four points, and linear fractional transformations produce an isomorphism . We try to make this into something resembling a functor, i.e., the projective line should be canonical. Take the sum-zero hyperplane , and equip this with the nondegenerate quadratic form . The zero set of this form defines a conic with four rational points, and we have an identification of points , where the zero is in the i-th coordinate. This induces an isomorphism . More generally, we get a diagram:
where the horizontal arrows are index two inclusions, and the vertical arrows are double cover maps. The central extension is the unique one for which transpositions lift to order two elements, and (2,2) permutations lift to order four elements.
At this point, Serre made a digression to talk about Wiles’ work on the Shimura-Taniyama conjecture. The sequence of ring maps induces a sequence of continuous group homomorphisms . There is an obstruction in to the existence of a section. Wiles was able to show that this obstruction vanished for the local Galois representations arising from elliptic curves. In fact, the section factors through . There is another interesting map that factors through , and this has something to do with the fact that thirty one is one more than the Coxeter number of while three is one more than the Coxeter number of , but I didn’t hear anything resembling an explanation.
Consider tree made from a central vertex of valence three, three outer vertices of valence three, and six vertices of valence one. What is its automorphism group? To examine it, we take a free rank two module over . The kernel of multiplication by two is a (2,2) group, with three nonidentity elements, which we identify with the outer branches. For each such element, we have two choices of a cyclic subgroup of order four that contains it, and we identify these with the six leaves. This action produces a surjection from to the automorphism group of this graph, with kernel . We have , where the first factor acts by switching all pairs of leaves. Where do we get the set of size four? Take the six leaves, and call them matched if they connect to the same outer branch. There are ways to split the six leaves into two classes of match representatives. Where does this come up? Take the 4-torsion points on an elliptic curve (over a field of characteristic not two). It is also the Weyl group of .
Consider a quartic extension of the rationals with symmetric or alternating Galois group. This produces a homomorphism from the absolute Galois group , and we can choose an index three subgroup and an index two subgroup . We can take the quotient to get a map , and this induces a transfer map . If we set to be the intersection of conjugates, we have . We get a tower of fields , where the first inclusion is cubic, and the second is quadratic, and there is some such that . The norm is rational, and it is a square if and only if the transfer is zero. Godwin has some tables of quartic fields.
Serre didn’t say much about sets of size five. We have . In particular, we get a natural action by automorphisms on . We also have , where is the group of semilinar transformations, i.e., a semidirect product with Galois automorphisms. This also naturally acts on the five points of by permutations. The action of is transitive on the six points of , so it induces a map to that is a composition of a standard inclusion with an outer automorphism. Unfortunately, Serre didn’t mention any canonical identifications of points like the previous cases, and I couldn’t find one by myself. The presence of semilinear transformations suggests the use of a restriction of scalars, but that doesn’t come out of a nice quadric.
We can view a six point set as the ramified points of the hyperelliptic quotient map from a smooth curve of genus two to . The two-torsion in the Jacobian of the genus two curve is a vector space of rank four over , and it comes with a natural symplectic form given by the Weil pairing. This provides an isomorphism .
Switching to characteristic three, we have , where and . There is a quadratic form on the four dimensional space , which defines a quadric surface in projective three-space. This surface has a rational point, and we can divide such quadrics into those with lines (isomorphic to ) and those without (isomorphic to the Weil restriction ). There is a subgroup of coordinate permutations that induces linear fractional transformations on , and in fact it realizes as the index two subgroup . The odd permutations correspond to semilinear transformations with square determinant, that is, they incorporate the Galois automorphism. The full group of semilinear transformations is , which contains as an index two subgroup, and this property is quite special. For all , there is a surjection , but for , there are outer automorphisms e.g. taking . One can prove this by enumerating transpositions. The other index two subgroups of are and the non-simple Mathieu group , which is not a split extension.
Serre didn’t say much about sets with seven elements, except to note that had a nontrivial central extension of order three, which is unusual. It can be found by studying the cohomology of the three-Sylow subgroup, and the action of the normalizer. This also works for , and the central extension is called the Valentine group, which has a three-dimensional complex representation, coming from a map . It fails for because it has an embedding of .
For sets of size eight, we can take spaces as before: , with a nondegenerate quadratic form on the six-dimensional quotient . This induces a map . has the same Dynkin diagram as , and indeed, . How do we see this? Take . The exterior square is six-dimensional, and has a split form from the map . Note that is a nonisomorphic simple group of the same order.
Serre also pointed out the exceptional isomorphism . One can take the modular curve , which is genus three, and isomorphic to the Klein quartic. Its automorphism group is , giving it a 168-fold cover of . Its Jacobian is three-dimensional, so its two-torsion points form a six-dimensional space over . This splits into a sum of two 3-dimensional representations under the action of automorphisms, inducing the exceptional isomorphism.