Soft planes and groups

Finite projective planes are constructed using groups that satisfy simple-looking conditions. The resulting projective planes include many known planes and possibly new ones, and are precisely those having a collineation group fixing a flag $(\mathrm{\infty },L_{\mathrm{\infty }})$ and transitive on the flags $(w,W)$ with $w\mathrm{\notin }{L}_{\mathrm{\infty }}$ and $\mathrm{\infty }\mathrm{\notin }W$.