SOFT-GROUPS AND SOFT-PLANES

Soft-groups (sort-of-flag-transitive groups) are automorphism groups of finite projective planes which have a large flag-orbit: specifically, they fix a distinguished flag and are transitive on all flags whose point is not on the distinguished line and whose line is not through the distinguished point. The planes they act upon in such a way are called soft-planes, and soft-planes can be constructed from any of their soft-groups. Since soft-groups feature a characterising structure, constructing new examples could be achievable using solely group-theoretical results, and this might lead to the discovery of new planes, however none has yet been found. We first cover the basics of finite geometry, then focus on the theorems which link soft-groups and soft-planes together, construct explicitly some of the few known examples, and finally prove a variety of group-theoretic properties of soft-groups using some theory of collineations.