Authors: Carl Droms, Brigitte Servatius and Herman Servatius

Reference: in Topology and Combinatorial Group Theory, Lecture Notes in Mathematics \#1440, Springer-Verlag, 52--59, 1990.

Abstract: A theorem of Marshall Hall, Jr.~\cite{Hall} (cf. also~\cite{Burns},~\cite{KarSolitar}) states that if $B = \{ h_{1},..., h_{k} \}$ is a free basis for a finitely generated subgroup $H$ of a f.g. free group $F$, and if $\{ x_{1},\ldots ,x_{n} \}$ is a finite subset of $F - H$, then $B$ can be extended to a free basis for a f.g. subgroup $H^{*}$ of finite index in $F$ such that $\{ x_{1},\ldots ,x_{n} \} \subset F - H^{*}$. A similar theorem holds for free abelian groups, and this may be regarded as a generalization of the fact that in a vector space, every linearly independent set can be extended to a basis. We will prove that an analogous {\em basis extension property} holds in a class of groups which contains the f.g. free and free abelian groups.