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