# On Schur Rings over Infinite Groups III

## TL;DR

This paper advances the understanding of Schur rings over infinite groups, especially over groups like d7a0p, providing structure theorems and classifications for special prime cases, and relating them to difference set partitions.

## Contribution

It offers new structure theorems for primitive sets and classifies all Schur rings over d7a0p for Fermat and safe primes as traditional.

## Key findings

- Complete classification of Schur rings over d7a0p for certain primes
- Structure theorems for primitive sets in these rings
- Analogies with difference set partitions

## Abstract

In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group $\Z\times\Z_p$, where $p$ is a prime. We provide structure theorems for primitive sets in these Schur rings. In the case of Fermat and safe primes, a complete classification theorem is proven, which states that all Schur rings over $\Z\times\Z_p$ are traditional. We also draw analogs between Schur rings over $\Z\times\Z_p$ and partitions of difference sets over $\Z_p$.