Discrete Laplace Cycles of Period Four

We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction

PGL(2, 11) and PSL(2, 11)

AbstractThese two groups both occur as projectivities on a finite line and as orthogonal transformations in a finite plane. The geometry of either of these two projective spaces provides interpretations of some known features of the groups, and each space provides two permutation representations of degree 11