This is really just a restatement of what Jim wrote, but here goes If you start with the (0,0,1), (1,1,0), etc labelling of 7 points in PG(2,2) in the "coat of arms" picture (ie " with obvious D_3 symmetry"), then the fact that there is also a element of order seven can be presented as a "surprising/shocking additional fact." Then back up to viewing PSL(2,7) as an group of symmetries on the points {1,2,...,7} that preserve the 2-(7,3,1) design and wave hands to argue all of these activities are really just the same thing. I agree that it's less than convincing somehow but I don't see how it can be any simpler. For the simplicity I would have to stoop to writing down matrices 1 0 a 1 and proving helper lemmas such as the statement that a normal subgroup of SL(2,F) containing such an element in fact has to be all of SL(2,F) ----- Original Message ----- From: "James Propp" <propp@math.wisc.edu> To: <math-fun@CS.Arizona.EDU> Sent: Friday, November 08, 2002 5:33 AM Subject: [math-fun] Re: PSL(2,F_7), SL(3,F_2), and the Fano plane
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun