#### Vol. 4, No. 1, 2006

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Stable planes with a point transitive abelian group

### Andreas Wörner

Vol. 4 (2006), No. 1, 53–61
##### Abstract

We consider stable planes $E=\left(P,\mathsc{ℒ}\right)$ 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 $\mathsc{ℒ}$ and use the results in order to characterize affine shift planes and translation planes.

##### Keywords
stable planes, abelian group action, arc planes
Primary: 51H10