[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
This is something that has intrigued me for long years. There's a Monthly article on it somewhere (between 10 & 20 years ago ???) If one knows how to search, one finds it. I have discussed it with JHC, who may respond -- I'll add his private address. R. On Fri, 8 Nov 2002, James Propp wrote:
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
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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
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participants (3)
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Thane Plambeck