Summary: 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.