Gaussian Binomial Coefficients in Group Theory, Field Theory, and Topology. Issue 5 (10th May 2022) - Chebolu, Sunil K.; Lockridge, Keir
Abstract: In this article we offer group-theoretic, field-theoretic, and topological interpretations of the Gaussian binomial coefficients and their sum. For a finite p -group G of rank n, we show that the Gaussian binomial coefficient ( n k ) p is the number of subgroups of G that are minimally expressible as an intersection of n – k maximal subgroups of G, and their sum is precisely the number of subgroups that are either G or an intersection of maximal subgroups of G . We provide a field-theoretic interpretation of these quantities through the lens of Galois theory and a topological interpretation involving covering spaces.