4.70 (1973)

Let $k$ be a field of characteristic different from 2, and $G_k$ the group of transformations $A = (a, \alpha) : x \mapsto ax + \alpha$, ($a, \alpha \in k$, $a \neq 0$). Extend $G_k$ to the projective plane by adjoining the symbols $(0, \alpha)$ and a line at infinity. Then the lines are just the centralizers $C_G(A)$ of elements $A \in G_k$ and their cosets. Do there exist other groups $G$ complementable to the projective plane such that the lines are just the cosets of the centralizers of elements of $G$?