Abstract

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

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Mathematical Subject Classification 2000

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