A characterization of \(SL_ 2(k)\) by its quadratic action on the natural module (Q1312354)

The author characterizes the group \(SL_ 2(k)\) by properties of its quadratic action on the natural module, where \(k\) is a skew field or a Cayley division algebra, compare \textit{F. Timmesfeld} [Forum Math. 6, No. 2, 209-231 (1994; Zbl 0802.51003)]. The assumptions on the action concern mainly a system of abelian subgroups, which turn out to be transvection subgroups. The proof uses geometric methods developed by the author and by \textit{J. I. Hall} [Proc. Lond. Math. Soc., III. Ser. 58, No. 1, 89-111 (1989; Zbl 0673.20015)] for the study of groups generated by \(k\)- transvection groups.