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

TL;DR

This paper characterizes when multiplicative subgroups of finite fields are generalized arithmetic progressions, showing they occur only at specific sizes, and extends additive decomposition results for these subgroups.

Contribution

It provides a complete characterization of multiplicative subgroups that are generalized arithmetic progressions and generalizes additive decomposition results to these subgroups.

Findings

Subgroups are GAPs only if their size is 2, 4, or p-1.

Additive n-decompositions of subgroups are necessarily direct sums.

Extends previous work on additive decompositions of subgroups.

Abstract

We prove that a multiplicative subgroup $A_k$ of $\mathbb{Z}_p^*$ 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 $\mathbb{Z}_p^*$, and we generalize a result of Hanson and Petridis to show that any additive $n$-decomposition of a subgroup must be a direct sum.