Free structure of factors

TL;DR

This paper investigates the free structure of factors in free groups, especially when the quotient is abelian, providing new results, methods, and applications including the construction of lower central factors and the development of relative basic commutators.

Contribution

It introduces new insights into the free structure of factors in free groups, develops the concept of relative basic commutators, and presents a collecting process for free generators with applications to group theory.

Findings

Precise description of free structure of Y relative to X when X/Y is abelian.

Development of the notion of relative basic commutators.

A collecting process for free generators applicable to subgroups.

Abstract

Factors $\frac{X}{Y}$ in a free group $F$ with $Y$ normal in $X$ are considered. Precise results on the free structure of ${Y}$ relative to the free structure of ${X}$ when $\frac{X}{Y}$ is abelian are obtained. Some extensions and applications are given, as for example to the construction of lower central factors in general groups. A collecting process on free generators, which gives basic commutator-type free generators for some subgroups, is also presented. The notion of {\em relative basic commutators} is developed.