Abelian group with elements finite orders.

Question

We know an abelian group has a subgroup which all elements in it has finite order . Now we remove condition abelian group , the clause is that true? If it's false give an example.

Answer 1

I think you just want torsion-free group: http://en.wikipedia.org/wiki/Torsion_%28algebra%29

Answer 2

A counter-example: the free product of two groups of the order 2 (more generally: of groups of finite orders).

Answer 3

Consider the trivial subgroup.