Coset Intersection Graphs for Groups

Let H,K be subgroups of G. We investigate the intersection properties of left and right cosets of these subgroups. If H and K are subgroups of G, then G can be partitioned as the disjoint union of all left cosets of H, as well as the disjoint union of all right cosets of K. But how do these two partitions of G intersect each other? Definition 1. Let G be a group, and H a subgroup of G. A left transversal for H in G is a set {tα}α∈I ⊆ G such that for each left coset gH there is precisely one α ∈ I satisfying tαH = gH. A right transversal for H in G in defined in an analogous fashion. A left-right transversal for H is a set S which is simultaneously a left transversal, and a right transversal, for H in G. A useful tool for studying the way left and right cosets interact, and obtaining transversals, is the coset intersection graph which we introduce here. Definition 2. Let G be a group and H,K subgroups of G. We define the coset intersection graph ΓH,K to be a graph with vertex set consisting of all left cosets of H ({liH}i∈I) together with all right cosets of K ({Krj}j∈J), where I, J are index sets. If a left coset of H and right coset of K correspond, they are still included twice. Edges (undirected) are included whenever any two of these cosets intersect, and the edge aH −Kb corresponds to the set aH ∩Kb. Observing that left (respectively, right) cosets do not intersect, we see that ΓH,K is a bipartite graph, split between {liH}i∈I and {Krj}j∈J . For H a finite index subgroup of G, the existence of a left-right transversal is well known, sometimes presented as an application of Hall’s marriage theorem [?]. When G is finite H will have size n, so any set