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It is well-known there is an isomorphism between $GL(3,2)=PGL(3,2)$, the automorphism group of the Fano plane (i.e. the projective plane over the finite field with two elements), and $PSL(2,7)$, which is the automorphism group of the oriented projective line over the field with seven elements. (More details are on Wikipedia).

What I'd like to know is if there is a finite geometric reason that these two groups are isomorphic. For instance, some combinatorial mapping of these geometries that induces an isomorphism between their automorphism groups. I was talking to Richard Green today about exceptional stuff in low dimensions and he claimed that there wasn't really a nice way to see it, unlike, for instance, the construction of the exceptional (outer) automorphism of $S_6$ using synthemes and duads.

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  • $\begingroup$ This article from AMM gives an elementary description of the isomorphism: math.vt.edu/people/brown/doc/PSL%282,7%29_GL%283,2%29.pdf (doi:10.4169/193009709X460859) but it's not geometric. It references a bunch of proofs, but I don't know if any of them are what I'm after. $\endgroup$ – David Roberts Dec 7 '16 at 6:40
  • $\begingroup$ @BjørnKjos-Hanssen that looks like it! $\endgroup$ – David Roberts Dec 7 '16 at 7:12
  • $\begingroup$ Even better, the comment math.stackexchange.com/questions/1401/… points out what might be the exact small numbers/low dimensions coincidence that allows such a construction. Namely, (8 choose k) divides 168 for k=0,1,2,3,5,6,7,8, but not k=4. $\endgroup$ – David Roberts Dec 7 '16 at 7:18
  • $\begingroup$ Or, more elementarily: $5\not\mid 168=2^3\cdot 3\cdot 7$ $\endgroup$ – David Roberts Dec 7 '16 at 7:36
  • $\begingroup$ How about explicitly transforming the 3x3 matrices over F_2 into 2x2 matrices over F_7 somehow... $\endgroup$ – Bjørn Kjos-Hanssen Dec 7 '16 at 7:44
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V. Dotsenko's construction, on math.stackexchange:

https://math.stackexchange.com/questions/1401/why-psl-3-mathbb-f-2-cong-psl-2-mathbb-f-7/1450#1450

may fit your requirement "combinatorial mapping of these geometries that induces an isomorphism".

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There is an explanation of sorts in Section 1.4 of Elkies's "The Klein quartic in number theory". There is a three-dimensional lattice $L$ over the cyclotomic field $k=\mathbf Q(\zeta_7)$, and $G$ can be defined as its group of isometries. The resulting three-dimensional representation of $G$ has the unusual property of remaining irreducible when reduced modulo every prime of $\mathcal O_k$. Its reduction modulo a prime over $2$ turns out to be $\mathrm{GL}(3,\mathbf F_2)$ acting on $\mathbf F_2^3$, and its reduction modulo a prime over $7$ is $\mathrm{PSL}(2,\mathbf F_7)$ acting on $\mathbf F_7^3$ as the symmetric square of the two-dimensional representation of $\mathrm{SL}(2,\mathbf F_7)$. (Note that since $-1$ acts trivially on the symmetric square, the symmetric square really is a projective representation.)

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  • $\begingroup$ I think 'of sorts' is the operative word. While there's some cool machinery in there, the bare combinatorics of the groups and their geometries as in the other answer beats out cyclotomic fields and number theory. $\endgroup$ – David Roberts Dec 7 '16 at 19:44

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