Three-factor decompositions of $\mathbb{U}_n$ with the three generators in arithmetic progression

TL;DR

This paper investigates when the multiplicative group of units modulo n can be decomposed into three cyclic groups generated by elements in arithmetic progression, revealing conditions and properties of such decompositions.

Contribution

It characterizes the conditions under which three-factor decompositions with generators in arithmetic progression exist for U_n, including for composite and prime power moduli.

Findings

Many decompositions exist for various n

Decompositions have additional interesting properties

Brief analysis of finite field multiplicative groups

Abstract

Irrespective of whether n is prime, prime power with exponent >1, or composite, the group U_n of units of Z_n can sometimes be obtained as the direct product of cyclic groups generated by x, x+k and x+2k, for x, k in Z_n. Indeed, for many values of n, many distinct 3-factor decompositions of this type exist. The circumstances in which such decompositions exist are examined. Many decompositions have additional interesting properties. We also look briefly at decompositions of the multiplicative groups of finite fields.