Let $G$ be a non-trivial torsion-free Abelian group.
If $T$ is a maximal free subgroup of $G$, then $G\over T$ is periodic?
How can we prove this?
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Show that if a coset $gT$ has infinite order in the factor group then $\langle g,T\rangle$ is a free-abelian subgroup of $G$ with proper subgroup $T$, contrary to the hypothesis on maximality of $T$.