2020/02/27 by Yuan, Pingzhi, Zhao, Kevin
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2002.11872
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 purely singular if for each prime divisor p of |G|, there is at least one element of M is divisible by p. In this paper, we mainly study the purely singular splittings of cyclic groups. We first prove that if k≥3 is a positive integer such that [-k+1, k]^* splits a cyclic group ℤm, then m=2k. Next, we have the following general result. Suppose M=[-k1, k2]^* splits ℤn(k1+k2)+1 with 1≤ k1< k2. If n≥ 2, then k1≤ n-2 and k2≤ 2n-5. Applying this result, we prove that if M=[-k1, k2]^* splits ℤm purely singularly, and either (i) gcd(s, m)=1 for all s∈ S or (ii) m=2αpβ or 2αp1p2 with α≥ 0, β≥ 1 and p, p1, p2 odd primes, then m=k1+k2+1 or k1=0 and m=k2+1 or 2k2+1.