An algorithm for finding minimal generating sets of finite groups

TL;DR

This paper introduces an algorithm leveraging Cayley graph components to find minimal generating sets of finite groups, connecting graph structure with subgroup generation.

Contribution

It presents a novel algorithm that constructs minimal generating sets of finite groups based on Cayley graph component analysis.

Findings

Algorithm effectively finds minimal generating sets for finite groups.

Cayley graph components relate to subgroup generation.

Provides a method to construct generating sets from graph components.

Abstract

In this article, we study connections between components of the Cayley graph $\mathrm{Cay}(G,A)$, where $A$ is an arbitrary subset of a group $G$, and cosets of the subgroup of $G$ generated by $A$. In particular, we show how to construct generating sets of $G$ if $\mathrm{Cay}(G,A)$ has finitely many components. Furthermore, we provide an algorithm for finding minimal generating sets of finite groups using their Cayley graphs.