Nonsingular splittings over finite fields

Abstract

Let G be a finite abelian group. We say that M and S form a splitting of G if every nonzero element g of G has a unique representation of the form g = ms with m ∈ M and s ∈ S, while 0 has no such representation. The splitting is called nonsingular if gcd(|G|, a) = 1 for any a ∈ 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 Zq, then there are infinitely many primes p such that M splits Zp.