Configuration sets for groups

Abstract

We develop practical tools for analyzing the configuration set $F(G,k)$ of $k\ge 2$ 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 $k\ge 2$ distinct non-identity elements in $G$.

16 pages. New title and author. Statements have been improved. Comments are welcome