On flag-transitive affine planes of order \(q^ 3\)
An affine plane is called flag-transitive if it admits a collineation group acting transitively on incident point-line pairs (or flags). By a celebrated result of \textit{A. Wagner} [Math. Z. 87, 1-11 (1965; Zbl 0131.368)] every finite flag-transitive plane is necessarily a translation plane. Recently several infinite families of flag-transitive planes with dimension \(n \geq 3\) over their kernels have been constructed [see the author, Graphs Comb. 7, 183-195 (1991; Zbl 0735.51005) and \textit{W. M. Kantor} and the author, J. Comb. Theory, Ser. A 65, No. 2, 307-310 (1994; Zbl 0801.51005)]. In the paper under review 3-dimensional flag-transitive planes are studied in some detail. Let \(\pi\) be a nondesarguesian translation plane of order \(q^ 3\) with kernel \(GF(q)\) where \(q\) is an odd prime power. Assume that \(G\) is a solvable collineation group of \(\pi\) in the linear translation complement which induces a transitive action on the line \(l_ \infty\) at infinity (hence \(\pi\) is flag-transitive). Let \(H\) be a maximal cyclic normal subgroup of \(G\). Then is is shown that \(H\) induces a semi-regular action with 1, 2, 3 or 6 orbits on \(l_ \infty\). Moreover, if \(q \not\equiv 2 \pmod 3\), then \(H\) induces at most 2 orbits on \(l_ \infty\) and \(\pi\) is one of the places mentioned in the above paragraph. Using an exhaustive computer search, it is shown that, up to isomorphism, the number of affine planes of order 27 (respectively, 125) admitting a solvable flag-transitive group with a cyclic subgroup which induces a semi-regular action of at most 2 orbits on \(l_ \infty\) is 3 (respectively, 5), one of which is the desarguesian plane. For \(q=3\) these are the only affine planes of order 27 admitting a solvable flag- transitive group. The author conjectures that for any odd prime \(q\not\equiv 2 \pmod 3\), the number of mutually nonisomorphic planes of order \(q^ 3\) admitting a solvable flag-transitive group is \(q\). In this regard it should be mentioned that the reviewer and R. D. Baker have recently shown that for any odd prime \(q\) the number of nonisomorphic affine planes of order \(q^ 2\) admitting a (solvable) flag-transitive group is \((q+1)/2\).
scientific article; zbMATH DE number 663754
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