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 $|A|\le e^{c(\log p{)}^{1/4}}$. In this paper, we will use a similar procedure to obtain an upper bound of the same type in the case of semidirect products ${\mathbb{Z}}_p{⋊}_{\phi }H$ where $\phi :H\to Aut({\mathbb{Z}}_p)$ satisfies $\phi (h)\in \{id,-id\}$ 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.