The Index Problem for Subgroup Intersections
Haran Mouli
TL;DR
This work reframes the index-problem for subgroup intersections as a graph-theoretic problem via coset intersection graphs, linking index-realizable triples to edge-transitive bipartite graphs with prescribed vertex counts. It establishes foundational necessary conditions, product-closure properties, and constructive realizations, including q-binomial coefficient based examples, while also proving non-realizability results that rule out large families of triples. The authors then apply these tools to classify all index-realizable triples with min(a,b) ≤ 10, illustrating the method's reach and limitations and drawing connections to compositum-feasible triples via Galois theory. Overall, the paper provides a concrete, graph-theoretic framework to tackle subgroup-intersection index problems and expands the catalog of explicitly realizable triples.
Abstract
In previous papers, Drungilas et al. study the problem of which triples of positive integers $(a, b, c)$ can be realized as $([E: \mathbb{Q}], [F: \mathbb{Q}], [EF: \mathbb{Q}])$, where $E$ and $F$ are number fields, using techniques from field theory. We shall study this problem rephrased in the language of groups using the Galois correspondence to simplify and generalize their results.