Transitive affine planes admitting elations
A long-standing problem in the theory of finite affine planes is the determination of those planes having a collineation group transitive on affine points. Such affine planes are called transitive affine planes. Several interesting results on this problem have been obtained [see the book ''Affine planes with transitive collineation groups'' by the reviewer [1982; Zbl 0485.51006]. Until 1966 it was conjectured that each such affine plane was a translation plane. In that year \textit{T. G. Ostrom} [Math. Z. 92, 201-209 (1966; Zbl 0178.234)] showed that those planes derived from the dual Lüneburg planes (of order \(2^{2r}\), where r is odd and \(r\geq 2)\) have a collineation group transitive on affine points, but they are not translation planes (nor dual translation planes). Since that time many such examples have been found: the affine planes derived from the dual Walker planes (of order \(q^ 2\) with \(q\equiv 5\) mod 6)), and the affine planes derived from the dual likeable planes [see \textit{W. Kantor}, Isr. J. Math. 42, 227-234 (1982; Zbl 0511.51011]. The above examples have led to a rephrasing of the original conjecture: Under what additional condition (or conditions) is a transitive affine plane a translation plane? The authors of the present article give one such condition: the plane admits (nontrivial) affine elations with at least two distinct centers. The proof depends on the following nice result: If a (finite) transitive affine plane admits a nontrivial affine elation with center \(A\in \ell\) then the plane is \((A,\ell_{\infty})\)- transitive. These results are then used to give the following characterization: If a transitive affine plane of order \(n\neq 81\) admits m nontrivial affine elations with distinct centers, where \(m\geq \sqrt{n}\), and the group generated by all affine elations is non-solvable then the plane is either desarguesian or Lüneburg-Tits.
scientific article; zbMATH DE number 3849969