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 their sum. For a finite p-group G rank n, show that coefficient (nk)p is number subgroups are minimally expressible as an intersection n – k maximal G, sum precisely either or G. We provide field-theoretic interpretation these quantities through lens Galois theory involving covering spaces.