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The objective of this note is to give some alternative proofs that the intersection of the normal subgroups with torsion-free nilpotent factor groups of free groups is the identity. Our approach gives rise to a new family of one-relator groups with this property. This family includes, among others, the fundamental groups of orientable surfaces. Our hope is that the idea involved will lead to new insights into these well-known groups, and, even more important for us, a proof that free $\mathbb{Q}$-groups have the same property.

A modification is suggested for a 1934 example of Reinhold Baer which shows that not all homomorphisms $C\to(\operatorname{Aut} A)/(\operatorname{Inn}\space A)$ arise from conjugation action in group extensions $1\to A\to B\to C\to 1.$