Graham's rearrangement for a class of semidirect products

Abstract

A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Bedert and Kravitz proved that this statement holds whenever $|A| \leq e^{c(\log p)^{1/4}}$. In this paper, we will use a similar procedure to obtain an upper bound of the same type in the case of semidirect products $\mathbb{Z}_p \rtimes_{\varphi} H$ where $\varphi: H \to Aut(\mathbb{Z}_p)$ satisfies $\varphi(h) \in \{id, -id\}$ for each $h \in H$ and where $H$ is abelian and each subset of $H$ can be ordered such that all of its partial products are distinct.