Why Is PSL(2, 7) ≅ GL(3, 2)?

1. INTRODUCTION. The groups of invertible matrices over finite fields are among the first groups we meet in a beginning course in modern algebra. Eventually, we find out about simple groups and that the unique simple group of order 168 has two representations as a group of matrices. And this is where we learn that the group of 2 × 2 unimodular matrices over a seven-element field, with I and −I identified, is isomorphic to the group of invertible 3 × 3 matrices over a 2-element field. In short, it is a fact that PSL(2, 7) ∼ = GL(3, 2). Many of us are surprised by this fact: why should a group of 2 × 2 matrices with mod-7 integer entries be isomorphic to a group of 3 × 3 binary matrices? There are a number of proofs of this remarkable theorem. Dickson [1, p. 303] gives a proof based on his general theorem giving uniform sets of generators and relations for the family of groups SL(2, q), where q is any prime power. One checks that the relations appearing in Dickson's presentation of PSL(2, 7) are satisfied by certain generators of GL(3, 2), implying that these groups have the same presentations and are therefore isomorphic. Dummit and Foote [2, p. 207–212] show that every simple group of order 168 is necessarily isomorphic to the automorphism group Aut(F) of the Fano plane F. They then show that Aut(F) ∼ = GL(3, 2) and that PSL(2, 7) is a simple group of order 168; the isomorphism theorem follows. Rotman gives the result as an exercise [5, Exercise 9.26, p. 281]. A hint is to begin with a simple group G of order 168 and use the seven conjugates of a Sylow 2-subgroup P of G to construct a seven-point projective plane; the proof is similar to Dummit and Foote's proof. Jeurissen [4] proves the result by showing that both PSL(2, 7) and GL(3, 2) are subgroups of index 2 of the automorphism group of a Coxeter graph. Elkies [3] gives a clever proof that uses the automorphism group G of the 3-(8, 4, 1) Steiner system—also known as the Steiner S(3, 4, 8) design. He shows that PSL(2, 7) is contained in G, which in turn maps homomorphically onto GL(3, 2). The result follows from the simplicity of the two groups and the fact that they are both of order 168. …