The Fano plane is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point.
So, What i want is to prove that if we have any other fano planes, they're all isomorphic.
Note : Please provide the answer without using the definition of groups. Try to prove it using graphs and not complex things.
What i tried was that i said every node is in 3 blocks. and somehow i should prove that in that 3 blocks, each way that i arrange the other 6 nodes, the new Fano plane is isomorphic to the last one.
Thanks in advance.
Hint: Assume we have Fano${}_1$ and Fano${}_2$ given, and pick and fix any three noncollinear points in both, call them e.g. $O_1,X_1,Y_1$ and $O_2,X_2,Y_2$. Try to express the lines and the further points in terms of 'intersections' and 'lines on two points', starting out from these three points.