Free structure of factors

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.