On Primitive Words I: A New Algorithm

TL;DR

This paper introduces a new elementary algorithm for free groups to determine subgroup relations and element primitivity, improving the efficiency of identifying free factors and primitive elements in free groups.

Contribution

It presents a novel, elementary algorithm for deciding if a subgroup is a free factor and if an element is primitive in free groups, advancing computational methods in group theory.

Findings

Algorithm effectively determines free factor relations.

Algorithm identifies primitive elements efficiently.

Applicable to subgroups of finite rank in free groups.

Abstract

Let $F_k$ be the free group on $k$ generators, and let $H\le J\le \F_k$ be subgroups of finite rank. We present a new elementary algorithm to determine whether $H$ is a free factor of $J$. In particular, this algorithm can determine whether a given element $w\in F_k$ is primitive, i.e. whether it belongs to some basis of $F_k$.