The Index Problem for Subgroup Intersections

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.