Talk:PSL(2,7)
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I have the nagging doubt that not "every automorphism of P1(7) is of this form", (in PSL(2,7)) since I seem to remember the symmetry group (group of projectivities) of P1(7) is PGL(2,7), not PSL(2,7). Of course, for P2(2), this doesn't matter, since PSL(3,2) = SL(3,2) = GL(3,2) = PGL(3,2). Revolver
If anyone is familiar with the actual isomorphism (or AN actual isomorphism) between PSL(2,7) and SL(3,2), this would be great. I did it as a homework problem a few years ago, but I don't remember all the details. Revolver
[edit] one of my favorite groups
It all depends what you mean by an automorphism of a projective line.
There are 336 transformations of the projective line over the field with 7 elements coming from the action of PGL(2,7), but only 168 coming from the PSL(2,7). The former group action is triply transitive, and indeed there are 8 x 7 x 6 = 336 triples of distinct points in this projective line. The latter group action is only transitive on oriented triples of points.
I would probably consider PGL(2,F) to be the 'full' symmetry group of the projective line over a field F, and PSL(2,F) to be the subgroup of orientation preserving symmetries.
I don't yet know an explicit isomorphism between PSL(2,7) and SL(3,2), but it's something I've been thinking about lately, so maybe I'll figure out a nice way to describe one.
John Baez 23:55, 23 Apr 2005 (UTC)
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