Sums of binomial coefficients modulo $p$ and groups of exponent $p^n$

TL;DR

This paper presents a matrix-based proof of binomial coefficient sum congruences modulo a prime p, and explores their application in constructing groups of specific exponents, shedding light on properties of the Heisenberg group.

Contribution

It introduces a simple matrix proof for binomial sum congruences and applies these to construct groups of exponent p^n, clarifying the role of prime p in group theory.

Findings

Provides a new proof technique for binomial sum congruences

Constructs groups of exponent p^n using these congruences

Explains the non-exceptional nature of p=2 in certain groups

Abstract

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as well as others of exponent $p^{n+1}$, explain why $p=2$ is not really an exceptional prime in relation to the Heisenberg group over the field with $p$ elements.