I don't know the answer, but it might be here: http://math.ucr.edu/home/baez/klein.html On Tue, May 22, 2012 at 1:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
The unique simple group of order 168 -- call it G_168 -- is the second-smallest nonabelian simple group (after A_5) and of special interest for a number of reasons.
It happens to be isomorphic to (==) both PSL(2,F_7) and PSL(3, F_2), abbreviated as PSL(2,7) and PSL(3,2).
This is one of only two non-trivial coincidences among the PSL(n, p^k) groups. (The other is that PSL(2,4) == PSL(2,5).)
QUESTION: What is the most direct proof that PSL(2,7) == PSL(3,2) ?
This almost seems like pure accident, since almost all PSL groups are simple and non-abelian, and there aren't many such groups of low order.
But maybe there is a proof that makes the isomorphism seem natural?
--Dan
P.S. One reason this group is interesting is that it's isomorphic to Aut(K), the group of conformal automorphisms of the surface of genus 3 having the largest automorphism group for its genus. This is the first example of a Riemann surface of genus g whose that attains the Hurwitz bound (for g >= 2) of 84(g-1).
It is realized by the Klein quartic K, the complex projective "curve" defined by the locus of roots of the homogeneous polynomial X^3 Y + Y^3 Z + Z^3 X, where [X:Y:Z] is the homogeneous coordinates of a point in the complex projective plane CP^2.
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