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

TL;DR

This paper explores the combinatorial, algebraic, and topological significance of Gaussian binomial coefficients, revealing their interpretations in group theory, field theory, and topology, especially in the context of subgroup structures.

Contribution

It provides novel interpretations of Gaussian binomial coefficients across group theory, Galois theory, and topology, connecting these areas through subgroup and covering space analysis.

Findings

Gaussian binomial coefficients count specific subgroup configurations in p-groups.

Sum of these coefficients corresponds to particular subgroup counts including the whole group.

Topological and field-theoretic interpretations offer new insights into subgroup enumeration.

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 $\binom{n}{k}_p$ 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