Multiplicative Subgroups of ${\mathbb{Z}}_p^{\ast }$ that are Generalized Arithmetic Progressions

Abstract

We prove that a multiplicative subgroup $A_k$ of ${\mathbb{Z}}_p^{\ast }$ is a generalized arithmetic progression if and only if $|A_k|=2,4,$ or $p-1$. Much of the argument is built upon recent work studying additive decompositions of subgroups of ${\mathbb{Z}}_p^{\ast }$, and we generalize a result of Hanson and Petridis to show that any additive $n$-decomposition of a subgroup must be a direct sum.