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The torsion subset T of G is the subset of G consisting of all elements that have finite order.

Let G be a finitely generated group with nontrivial finite derived subgroup.is the torsion subset T form a subgroup of G?

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    Well it's true in $G/G'$, and $G'$ is finite.2012-11-08

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It works even more generally: if G' is included in T, then T is a subgroup of G.

And so, if T is not a subgroup of G then G has at least one commutator of infinite order.