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Let G be an abelian group, $T$ the torsion subgroup of $G$. If $G/T$ is torsion-free, then $T$ and $G/T$ must be disjoint. $G=T \bigoplus G/T$ implies this as well. I don't understand why they are disjoint in the first place.

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    This is a mess: $T$ and $G/T$ don't even live in the same group. And aren't you assuming that $G/T$ is finitely generated or something?2012-04-26

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