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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 the torsion subset T form a subgroup of G?

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What is a derived subgroup? – Rasmus Nov 8 '12 at 14:23
What have you tried? Do you know the answer if the derived subgroup is trivial? – jspecter Nov 8 '12 at 14:23
The derived subgroup of a group is the subgroup generated by all the commutators of the group. – mojtaba farazi Nov 8 '12 at 14:27
If the derived subgroup be trivial then G is abelian group and clearly T is a form subgroup of G. – mojtaba farazi Nov 8 '12 at 14:29
Well it's true in $G/G'$, and $G'$ is finite. – Derek Holt Nov 8 '12 at 16:11

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.

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