On collineation groups with block orbits
During the 1960's and 1970's finite affine planes \(\Pi\) having collineation groups \(G\) transitive on the affine points were investigated, and theorems characterizing such planes were obtained. In many cases -- for example, planes with groups doubly transitive on the affine points -- the crux of the matter was the fact that the subgroup \(G_ 0\) fixing an affine point 0 has a block orbit; i.e., a point orbit \(\Gamma\) such that \(\Gamma \cup \{0\}\) is the union of one or more affine lines through 0 [see the reviewer's book `Affine planes with transitive collineation groups' (1982; Zbl 0485.51006)]. In that text the reviewer made the conjecture that if \(G_ 0\) has a block orbit \(\Gamma\) with \(\Gamma \cup \{0\}\) the union of at least two lines, then \(\Pi\) is a translation plane and \(G\) contains the group of translation. \textit{N. L. Johnson} and the reviewer [Math. Z. 135, 149-164 (1974; Zbl 0274.50017)] showed that the conjecture is true under any one of additional conditions -- for example, if the plane has nonsquare order, or if its order is even. In the present article the author shows that the conjecture is true without any additional hypothesis. The reviewer also made two other conjectures: (i) If \(\Gamma \cup \{0\}\) is an affine line through 0 then \(\Pi\) is either a translation plane or a dual translation plane, and the group \(G\) contains the translations or dual translations, respectively. (ii) If \(G_ 0\) has two block orbits then \(\Pi\) is a translation plane and \(G\) contains the translations. The author gives an example of a collineation group \(G\) on a desarguesian plane which satisfies the hypothesis of (i) and (ii), but does not satisfy the conclusions regarding \(G\). This example also disproves a conjecture of Johnson that every block orbit is a strong block orbit. Note, however, that the example does not disprove the conclusions in (i) and (ii) regarding the plane \(\Pi\), but it makes them less likely to be true.
scientific article; zbMATH DE number 446860