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Configuration sets for groups

Cesar A. Ipanaque Zapata, Alex Freitas de Campos

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 $G$.

Configuration sets for groups

Abstract

We develop practical tools for analyzing the configuration set of distinct elements in a group . We apply our results to design homogeneous linear systems in that admit nontrivial solutions. Furthermore, we study the connectivity of Cayley graphs of the form . In addition, we consider the configuration set of distinct non-identity elements in .

Paper Structure

This paper contains 7 sections, 11 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. The configuration set $F(G,k)$
  3. Applications
  4. Characterization of $F(G,k)$
  5. Linear systems in $\mathbb{F}_q$
  6. Cayley graphs
  7. The set $F(G\setminus\{1\},k)$

Key Result

Proposition 2.3

Let $G$ be a group and $k\geq 3$ such that $|G|\geq k$. We have $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (35)

  • Remark 1.1
  • Example 2.1
  • Definition 2.2: Norm
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 25 more