Finite Simple Groups and Dickson Invariants

In a recent paper [1], Adem, Maginnis and Milgram calculated the mod 2 cohomology of the Mathieu simple group M12. An interesting feature of the answer they obtain is that it is Cohen–Macaulay. There is a subalgebra isomorphic to the rank three Dickson invariants, namely a polynomial ring in generators of degrees 4, 6 and 7, and over this subalgebra the whole cohomology ring is a finitely generated free module. Since the classifying space BG2 of the compact Lie group G2 has the rank three Dickson invariants as its cohomology, one is led to ask whether there is an interesting relationship between M12 and G2. There is clearly no group homomorphism from M12 to G2 which induces the inclusion of the Dickson invariants into the cohomology of BM12. Indeed, there is no non-trivial group homomorphism from M12 to G2 at all, since the smallest nontrivial complex representation of M12 has dimension 11, while G2 has a 7 dimensional complex representation. Nevertheless, as we shall see in Section 5, there is a map of classifying spaces with the desired properties.