# The exceptional isomorphism between PGL(3,2) and PSL(2,7): geometric origin?

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.

• 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. – David Roberts Dec 7 '16 at 6:40
• @BjørnKjos-Hanssen that looks like it! – David Roberts Dec 7 '16 at 7:12
• 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. – David Roberts Dec 7 '16 at 7:18
• Or, more elementarily: $5\not\mid 168=2^3\cdot 3\cdot 7$ – David Roberts Dec 7 '16 at 7:36
• How about explicitly transforming the 3x3 matrices over F_2 into 2x2 matrices over F_7 somehow... – Bjørn Kjos-Hanssen Dec 7 '16 at 7:44

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.)