Automorphisms of finite projective planes

Please consult the handout about Finite Projective Planes.

Conjecture 3.4   Not every finite group is isomorphic to the group of automorphisms (collineations) of a finite projective plane. In fact, not every finite group is isomorphic even to a subgroup of the automorphism group of a finite projective plane. $A_6$ appears to be a candidate; $A_{100}$ looks like an easier one to rule out as a subgroup.

Conjecture 3.5   If $P$ is a projective plane of order $n$, then $\vert\mathop{\rm Aut}(P)\vert\le n^C$ for some constant $C$. Perhaps, $\vert\mathop{\rm Aut}(P)\vert\le n^{8+o(1)}$.