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 $\binom{n}{k}_p$ 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