Nonsingular splittings over finite fields

TL;DR

This paper investigates nonsingular splittings of cyclic groups over finite fields, introducing a new notation and establishing conditions for the existence of such splittings across infinitely many primes.

Contribution

The paper introduces the concept of direct KM logarithm and proves the existence of infinitely many primes where a given splitting occurs.

Findings

If a prime q exists such that M splits Z_q, then infinitely many primes p exist where M splits Z_p.

Introduction of direct KM logarithm notation for analyzing splittings.

Establishment of conditions linking splittings over different primes.

Abstract

We say that $M$ and $S$ form a \textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$, while $0$ has no such representation. The splitting is called {\it nonsingular} if $\gcd(|G|, a) = 1$ for any $a\in M$. In this paper, we focus our study on nonsingular splittings of cyclic groups. We introduce a new notation --direct KM logarithm and we prove that if there is a prime $q$ such that $M$ splits $\mathbb{Z}_q$, then there are infinitely many primes $p$ such that $M$ splits $\mathbb{Z}_p$.