A remark on finite point transitive affine planes with two orbits on \(l_ \infty\) (Q1802979)

In his previous work [ibid. 27, No. 2, 271-276 (1990; Zbl 0705.51002)] the author proved the following Kallaher's conjecture [\textit{M. J. Kallaher}, ``Affine planes with transitive collineations groups'' (1982; Zbl 0485.51006)]: Let \(\pi\) be an affine plane of odd order, \(n\) admits a collineation group \(G\) that acts transitively on its affine points and has two orbits of lengths \(n-1\) and 2 on \(l_ \infty\), then \(\pi\) is a translation plane. At present he spreads the result meant above: Assume \(| G(l_ \infty,l_ \infty)|=n=2^ m\) for some \(m\geq 1\), \(G(p_ 1,l_ \infty)=G(p_ 2,l_ \infty)=1\) and \(| G(p,l_ \infty)|=2\) for all \(p\in l_ \infty-\{p_ 1,p_ 2\}\), where \(p_ 1,p_ 2\) is a \(G\)- orbit of length 2 on \(l_ \infty\). Then \(n=2\) and \(G\) is a cyclic group of order 4. Remark: This theorem has been proved earlier by H. B. Maharjan under the condition that \(n\leq 4\).