On Sequences in Cyclic Groups with Distinct Partial Sums

Abstract

A subset of an abelian group is sequenceable if there is an ordering (x 1 , . . . , x k ) of its elements such that the partial sums (y 0 , y 1 , . . . , y k ), given by y 0 = 0 and y i = i j=1 x i for 1 ≤ i ≤ k, are distinct, with the possible exception that we may have y k = y 0 = 0. We demonstrate the sequenceability of subsets of size k of Z n \ {0} when n = mt in many cases, including when m is either prime or has all prime factors larger than k!/2 for k ≤ 11 and t ≤ 5 and for k = 12 and t ≤ 4. We obtain similar, but partial, results for 13 ≤ k ≤ 15. This represents progress on a variety of questions and conjectures in the literature concerning the sequenceability of subsets of abelian groups, which we combine and summarize into the conjecture that if a subset of an abelian group does not contain 0 then it is sequenceable.

Cycle type in Hall-Paige: A proof of the Friedlander-Gordon-Tannenbaum conjecture

“…It turns out that even for cyclic groups of prime order, this is an open problem, posed initially by Ronald Graham in 1971. Surprisingly little is known about this problem (see Problem 10 from [21], see also [4,13,12,11]). For example, the problem is already open for k = 13 and cyclic groups of prime order.…”