Intrinsic Geometries of the Simple Group of Order 168.

It is well known that, up to isomorphism, there is a unique simple group of order 168. The projective special linear groups PSL(2, 7) and PSL(3, 2) are concrete examples of such groups, and are therefore isomorphic. This isomorphism is not obvious, but it has been clarified from several viewpoints. Here we use purely group-theoretic methods—the Sylow theorems, Poincaré's theorem, and the orbit-stabilizer theorem—to show that any simple group of order 168 has a natural action on the projective line over F 7 , so it is isomorphic to PSL(2, 7). We discuss an analogous action on the projective plane over F 2 described by Smith and Tabachnikova. Both geometries are constructed using conjugacy classes of maximal subgroups. This gives another proof of the uniqueness theorem and may dispel some of the air of mystery that often surrounds the isomorphism between PSL(2, 7) and PSL(3, 2). [ABSTRACT FROM AUTHOR]