A little while ago, Gordon wrote about a conjecture about projective planes whose automorphism group acts transitively on points. It is thought that the Desarguesian planes are the only examples.

The best results on groups acting on projective planes in the last five years are due solely to Nick Gill, and he has recently posted a preprint to the arxiv containing an astonishing result. Gill’s work seems to have not gained the profile it deserves and so we want to make special mention of his main theorems (n.b., $O(G)$ is the largest odd-order normal subgroup of $G$):

Theorem A: Suppose that a group $G$ acts transitively on the set of points of a finite non-Desarguesian projective plane. Then the rank of the largest elementary-abelian 2-subgroup of $G$ is at most $1$.

Theorem B: Suppose that a group $G$ acts transitively on the set of points of a non-Desarguesian projective plane. Then we have the following possibilities:

1. $G$ has a normal 2-complement (and so $G$ is soluble).
2. $G/O(G)$ is isomorphic to $SL_2(3)$ or to a non-split degree 2 extension
of $SL_2(3)$ (and so $G$ is soluble).
3. $G = O(G) : SL_2(5)$ or $G = (O(G) : (SL_2(5)).2$

So to summarise, if a group $G$ acts transitively on a finite non-Desarguesian projective plane, then the Sylow 2-subgroups of G are cyclic or generalized quaternion, and if $G$ is insoluble, then $G/O(G)$ is isomorphic to $SL_2(5)$ or $SL_2(5).2$.

Well done Nick!