Stable planes with a point transitive abelian group

Abstract

We consider stable planes $E=\left(P,\mathcal{ℒ}\right)$ admitting an abelian group $\mathrm{\Delta }$ which acts (sharply) transitively on the point set $P$. Known examples are the so-called arc planes with $\mathrm{\Delta }={ℝ}^2$ (H. Groh, 1979), affine translation planes and shift planes. We examine the possible actions of $\mathrm{\Delta }$ on the line space $\mathcal{ℒ}$ and use the results in order to characterize affine shift planes and translation planes.