Adding generators in cyclic groups

TL;DR

This paper investigates the structure of sumsets formed by atoms in finite cyclic groups, providing detailed descriptions of their composition and element representations.

Contribution

It offers a detailed analysis of how atoms in cyclic groups combine, including characterization of the sumset structure and enumeration of element representations.

Findings

Sumsets of atoms are disjoint unions of atoms.

The paper characterizes which atoms form the sum of two given atoms.

It computes the number of representations for each element in the sumset.

Abstract

For a cyclic group $a$, define the atom of $a$ as the set of all elements generating $a$. Given any two elements $a,b$ of a finite cyclic group $G$, we study the sumset of the atom of $a$ and the atom of $b$. It is known that such a sumset is a disjoint union of atoms. The goal of this paper is to offer a deeper understanding of this phenomenon, by determining which atoms make up the sum of two given atoms and by computing the exact number of representations of each element of the sumset.