Nilpotent Singer groups

Gill, Nick (2006). Nilpotent Singer groups. Electronic Journal of Combinatorics, 13(1) R94.

URL: http://www.combinatorics.org/Volume_13/PDF/v13i1r9...

Abstract

Let $N$ be a nilpotent group normal in a group $G$ . Suppose that $G$ acts transitively upon the points of a finite non-Desarguesian projective plane $\mathcal{P}$ . We prove that, if $\mathcal{P}$ has square order, then $N$ must act semi-regularly on $\mathcal{P}$ .
In addition we prove that if a finite non-Desarguesian projective plane $\mathcal{P}$ admits more than one nilpotent group which is regular on the points of $\mathcal{P}$ then $\mathcal{P}$ has non-square order and the automorphism group of $\mathcal{P}$ has odd order.