Coset Intersection Graphs for Groups

Abstract

Abstract. 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 {t α } α∈I ⊆ G such that for each left coset gH there is precisely one α ∈ I satisfying t α H = 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 Γ G H,K to be a graph with vertex set consisting of all left cosets of H ({l i H} i∈I ) together with all right cosets of K ({Kr j } 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 Γ G H,K is a bipartite graph, split between {l i H} i∈I and {Kr j } 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 [3]. When G is finite H will have size n, so any set of k left cosets of H intersects at least k right cosets of H (or their union would have size < kn).

Transversals as Generating Sets in Finitely Generated Groups

“…However, it is not obvious that a left-right transversal always exists. We gave a short proof of this in [3] for the case where H is of finite index, as well as a brief historical discussion of this result.…”

“…This requires a deeper understanding of how cosets intersect, as discussed in Section 2. We urge the reader to consider the discussion of 'chessboards' given after Corollary 2.3, and to consult [3] for an example. These are vital in proving what follows.…”

Investigating transversals as generating sets for groups

“…It turns out that such a transversal always exists, by an application (see [2]) of Hall's Marriage Theorem to the coset intersection graph Γ G H,H of H in G (Definition 2.1); a graph whose vertex set is the disjoint union of the left and the right cosets of H in G, with edges between vertices whenever the corresponding cosets intersect. Γ G H,H is thus bipartite as the set of left (and right) cosets are mutually disjoint.…”

“…To achieve this, the authors introduced a new technique called shifting boxes. This involves using the transitive action of a group G on the set of left (or right) cosets of a subgroup H G to apply Nielsen transformations to a generating set of G in a way such that the resulting generators lie inside (or outside) particular desired cosets of H. A study of the graph Γ G H,H was conducted in [2], and it was shown that the components are always complete bipartite graphs. This description of Γ G H,H gave rise to a combinatorial model of coset intersections known as 'chessboards'.…”

“…In Section 2 we give an overview of the structure results and techniques introduced in [2,3] on coset intersection graphs and chessboards. In Section 3 we generalise some of the shifting boxes techniques from [3] to obtain further results which become our main approach to answering questions about leftright transversals as generating sets.…”