Gaussian Binomial Coefficients in Group Theory, Field Theory, and Topology. Issue 5 (10th May 2022)

• Record Type: Journal Article
• Authors: Chebolu, Sunil K.; Lockridge, Keir
• Is Part Of: American Mathematical Monthly. Volume 129:Issue 5(2022)
• Page Start: 466
• Page End: 473
• Publication Date: 2022-05-10
• Subjects: Primary 06A15 -- Secondary 20D15 -- 57M10; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2022.2040320
• Languages: English
• ISSNs: 0002-9890

Abstract

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.