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

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 ${(\begin{array}{c}n\\ k\end{array})}_p$ 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.

MSC:

• Primary 06A15
• Secondary 20D15
• 57M10

Acknowledgment

We thank Jon Carlson for providing helpful feedback on an early version of this article. We also thank Andy Schultz and Craig Westerland for their suggestions.

Additional information

Notes on contributors

Sunil K. Chebolu

Sunil Chebolu received his Ph.D. in mathematics from the University of Washington in 2005. He is currently a professor at Illinois State University, where he does research in algebra and number theory. In his spare time, he enjoys playing his guitar or observing deep-sky objects through his telescope.

Keir Lockridge

Keir Lockridge received his Ph.D. in mathematics from the University of Washington in 2006, and he now teaches at Gettysburg College. His primary mathematical interests include ring theory, number theory, and topology. In his spare time, he enjoys experimenting with fermentation or going on other culinary adventures.