Geometries from Groups
Hausdorff, metric geometry, betweenness, group, subgroup, word length, finite geometries
Koch Larue, Casey D., "Geometries from Groups" (2017). Student Summer Scholars. 185.
In this paper we describe a new technique for geometrically representing the subgroup structure of a finite group. The approach is to create a finite geometry based on a metric on the set of subgroups of a group, which in turn comes from a metric defined using word lengths corresponding to a given generating set for the same group. Going from a metric on a group to a metric on the set of subgroups of the same group is accomplished using the Hausdorff metric. Finite geometries emerge in which the points are the subgroups of a finite group and the lines are sets of subgroups of the same finite group, where lines are determined by a notion of betweenness based on the Hausdorff metric. We introduce a notion of embeddings of geometries, and we observe that the geometry for the quotient of an abelian group embeds the geometry for the original group. Also, we provide a complete characterization of the lines in the geometries for the finite cyclic groups with the standard generator. Applying this result, we investigate the way in which the geometry for the cyclic group of order n embed the geometries for cyclic groups with an order of a multiple of n. We finally determine and showcase the geometries that have 5, 6, or 7 points and arise from groups with respect to standard choices of generating sets. This research was conducted as part of the 2017 Student Summer Scholars Program at Grand Valley State University.