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?)
Best Answer
The standard way to prove this is using conjugacy classes as demonstrated here.
But here is an amusing geometric/combinatorial proof of the order.
Define $PG(n, p^m)$ as the geometry defined by the subspaces of $V (n + 1, p^m)$, an $n+1$ dimensional vector space over $F_{p^m}$; we call the 1 dimensional subspaces points. Recall that a Projective Geometry of the form $PG(2, p^m)$ has the following properties:
1) Every two points are incident with a unique line.
2) Every two lines are incident with a unique point.
3) There exist a set of four points, such that any 3 element subset is not collinear.
And notice that $PG(2, 2^1)$ is the Fano Plane. Note that $F_2$ is of characteristic 2.
Lemma: Three points such that all three are not collinear, uniquely determine the Fano plane.
proof: Let $a, b, c \in D$, we wish to determine the seven lines of our diagram. By the axioms of projective planes, we know that every two points are collinear and since each line has three points the linear combination; $a + b, a + c, b + c$ complete three lines. By our assumption, $a, b, c$ are not collinear so these points are all distinct. By the axioms of projective planes we know that $a + b$ is incident on $3$ lines. Observe that ${a + b, a + c, b + c}$ are collinear: $(a + b)+(a + c) = a+b+a+c = 2a+b+c = b+c$ in characteristic 2. Thus $a + b$ is incident with another line. Consider the line composed of the points ${a + b, c, (a + b) + c}$, then we have determined our diagram.
Proposition: The automorphism group of the Fano plane is order 168.
proof There are 168 choices of three points which are not collinear.
You can use a proof similar to the above to show the simplicity of this automorphism group. Then you could prove that the only simple group of order 168 is $PSL(2,7)$.