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In finite geometry, the Fano plane is the finite projective plane with the smallest possible number of points and lines: 7 each.

Why its automorphism group is $PSL(2,\mathbb{F}_7)$?

For the platonic solids, their automorphism group (orientation preserving) are $A_4, S_4$, and $A_5$. For the Fano plane, can we consider automorphisms which preserve orientation? (i.e. is the group $PSL(2,\mathbb{F}_7)$ orientation preserving?)

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    What do you mean by "orientation" of the Fano plane? Notice that the automorphism group is simple, so it cannot act transitively on any set of two elements: if there are two of whatever it is you mean by "orientation", then all automorphism must preserve each of them---and if there are more than two orientations in the sense you have in mind, well, then you have in mind something funny :)2011-04-21
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    I think it's much more natural to show that the automorphism group is $\mbox{PSL}(3,\mathbb{F}_2)$. You then need to convince yourself that it's the same group...2011-04-21
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    See also http://math.stackexchange.com/questions/1401/why-psl-3-mathbb-f-2-cong-psl-2-mathbb-f-7 (possible duplicate?)2011-04-21

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