Configuration sets for groups

TL;DR

This paper introduces tools for analyzing configuration sets in groups, with applications to linear systems over finite fields and Cayley graph connectivity.

Contribution

It provides practical methods for studying configuration sets in groups and explores their applications in linear systems and graph connectivity.

Findings

Developed tools for analyzing configuration sets in groups.

Applied results to design linear systems with nontrivial solutions.

Studied connectivity properties of Cayley graphs.

Abstract

We develop practical tools for analyzing the configuration set $F(G,k)$ of $k\geq 2$ distinct elements in a group $G$. We apply our results to design homogeneous linear systems in $\mathbb{F}_q$ that admit nontrivial solutions. Furthermore, we study the connectivity of Cayley graphs of the form $\mathrm{Cay}(G^k,F(G,k))$. In addition, we consider the configuration set of $k\geq 2$ distinct non-identity elements in $G$.