Nilpotent Singer groups

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



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.

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