Talk:PSL(2,7)

Some links

http://mathforum.org/kb/message.jspa?messageID=227628 - min. pols. taken from here

http://math.berkeley.edu/~dpenneys/math252/cueto_16_8.pdf character table

http://crazyproject.wordpress.com/2011/11/08/exhibit-a-set-of-representatives-of-the-conjugacy-classes-of-gl3-zz2/ representative of conj. classes

--Alexander Chervov (talk) 19:46, 11 August 2012 (UTC)[reply]

Not every automorphism

Title added. —Nils von Barth (nbarth) (talk) 04:23, 16 April 2010 (UTC)[reply]

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

Thanks! You are correct – PGL(2,7) acts (sharply) 3-transitively one the projective line (as the order suggests: 8*7*6=336), while PSL(2,7) does not, as the order indicates. Further, some people talk about collineations, so I’ve clarified this is the text.

—Nils von Barth (nbarth) (talk) 04:23, 16 April 2010 (UTC)[reply]

Hey, Nils! Just reading up on PSL(3,2). simon 192.12.12.217 (talk) 02:18, 8 June 2011 (UTC)[reply]

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)

Dead link

During several automated bot runs the following external link was found to be unavailable. Please check if the link is in fact down and fix or remove it in that case!

http://www.innerx.net/personal/tsmith/cdomain.html
- In PSL(2,7) on Mon Jul 17 14:36:09 2006, Socket Error: (-3, 'Temporary failure in name resolution')
- In PSL(2,7) on Thu Jul 27 00:34:07 2006, Socket Error: (-3, 'Temporary failure in name resolution')

maru (talk) contribs 04:34, 27 July 2006 (UTC)[reply]

Cayley graph

I've just drawn a Cayley graph for this group, it's at http://www.weddslist.com/groups/genus3/psl(2,7).html. Unfortunately it's too big to fit into an article sensibly. Maproom (talk) 22:57, 9 May 2009 (UTC)[reply]

Wrong argument

There was an affirmation that 21 does not divide 168 (=8*21) as a justification for it being not transitive on 21 points! The consequence is true in this case but not the argument. Ogerard (talk) 18:48, 28 September 2011 (UTC)[reply]

The isomorphism between PSL(2,7) and PSL(3,2) should be described

One of the most fascinating things about PSL(2,7) and PSL(3,2) is that the two groups are isomorphic -- a very rare occurrence in the world of PSL's. (There is only one other nontrivial coincidence: PSL(2,4) ≈ PSL(2,5), both being isomorphic to A₅.)

This fact that PSL(2,7) ≈ PSL(3,2) is fortunately mentioned in the article, though not elaborated on. It seems weirdly asymmetrical to me that the name of the article is PSL(2,7) but the isomorphic group PSL(3,2) is almost not mentioned at all, and the isomorphism between them is not described.

I am not an expert in group theory, but someone who is might try their hand at describing an isomorphism between these two groups — especially one of the most natural isomorphisms, if such exist. (I also believe the two groups should be treated more or less symmetrically in the article, and the name of the article should ideally be changed to PSL(2,7) and PSL(3,2). Or at minimum any search for PSL(3,2) should be redirected here.)Daqu (talk) 20:06, 3 May 2012 (UTC)[reply]

A good source would be Brown, Ezra; Loehr, Nicholas (2009). "Why is PSL (2,7)≅ GL (3,2)?" (PDF). Am. Math. Mon. 116 (8): 727–732. doi:10.4169/193009709X460859. Zbl 1229.20046. Deltahedron (talk) 07:07, 27 September 2014 (UTC)[reply]

D. Richter's web page describing a "polyhedral immersion" of the triangular tiling of the Klein quartic is erroneous

D. Richter claims on his web page http://homepages.wmich.edu/~drichter/mathieu.htm (and since when do we use web pages as references????? They haven't been peer-reviewed) that the tiling by 56 triangles of the Klein quartic can be polyhedrally immersed in 3-space via the small cubicuboctahedron. But this is wrong, as I have edited the article on that polyhedron to reflect: it is not immersed in 3-space because it contains singularities that are cones on the figure-8. And so I have removed the article's mention of this erroneous claim.

Although one can correct the statement to make it accurate, the corrected statement is so tenuously related, if at all, to the subject PSL(2,7) of this article that it is a far better thing to omit it entirely.Daqu (talk) 14:20, 16 August 2012 (UTC)[reply]