# Difference bases in finite Abelian groups

## TL;DR

This paper investigates the minimal size of difference bases in finite Abelian groups, establishing recursive bounds and explicit calculations for groups of small order, advancing understanding of their algebraic structure.

## Contribution

It introduces recursive upper bounds for difference sizes and characteristics of finite Abelian groups using Galois rings, and computes exact difference sizes for groups under 96 elements.

## Key findings

- For prime p ≥ 11, difference characteristic < (√p - 1)/(√p - 3) times the supremum over cyclic p-groups.
- Derived recursive bounds for difference sizes and characteristics of finite Abelian groups.
- Calculated difference sizes for all Abelian groups with fewer than 96 elements.

## Abstract

A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $\Delta[G]$. The fraction $\eth[G]:=\frac{\Delta[G]}{\sqrt{|G|}}$ is called the difference characteristic of $G$. Using properies of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $p\ge 11$, any finite Abelian $p$-group $G$ has difference characteristic $\eth[G]<\frac{\sqrt{p}-1}{\sqrt{p}-3}\cdot\sup_{k\in\mathbb N}\eth[C_{p^k}]<\sqrt{2}\cdot\frac{\sqrt{p}-1}{\sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality $<96$.