I asked:
What's a good way to see that PSL(2,F_7) is isomorphic to SL(3,F_2)?
"Good" is of course a subjective term; I was trying to come up with someething suitable for a class that I'm teaching. One slick way might be to 1) give two models of the projective plane over F_2 (one with the usual D_3 symmetry, the other given by translates of {1,2,4} in F_7, with 7-fold rotational symmetry), 2) show (non-constructively) that the 7-point finite projective plane is unique (so that the two models HAVE to be the same), 3) show that the symmetries of the first are given by SL(3,F_2), 4) show that the symmetries of the second are given by PSL(2,F_7), 5) ask them to explicitly construct an element of SL(3,F_2) of order 7. Of course, this approach leaves one with a lingering sense of mystery. Is there a good way to see that the two models of the 7-point finite projective plane are the same? I'd like to wrap things up with a proof of simplicity (especially if there's a sweet proof that takes advantage of the fact that the group has these two different representations). Can anyone think of one? (Is John Conway still reading math-fun? I'd've expected him to know a two or three answers to each of my questions off the top of his head. It seems he hasn't posted since late September.) Jim Propp