Gaussian Binomial Coefficients in Group Theory, Field Theory, and Topology

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.