Über schiefaffine Räume der Ordnung 2. (On skew-affine spaces of order two) (Q1096169)

By allowing the operation of joining two points to be noncommutative, the author has generalized affine spaces to skew-affine spaces, cp. also \textit{J. Pfalzgraf} [J. Geom. 25, 147-163 (1985; Zbl 0581.51008)]. Finite tactical skew-affine spaces of order \(k\geq 3\) (i.e. every line has exactly k points) have been investigated in the author's papers in Math. Z. 154, 159-168 (1977; Zbl 0336.05017) and Abh. Math. Semin. Univ. Hamb. 51, 120-135 (1981; Zbl 0474.51006). Here the author shows that tactical skew-affine spaces R of order 2 correspond bijectively to sharply transitive permutation groups G (more generally, tactical spaces of order 2 correspond to sharply transitive permutation sets). Furthermore, R satisfies the parallelogram condition if and only if G is abelian, and the lines of R are determined uniquely by their ``proper'' parts if and only if R is an affine space over GF(2) (and G is elementary abelian of exponent 2).