Yes. I mean, if you had with then in both and . I amn't even sure how to explain this...it is kinda fundamental...! Basically, if is a subgroup, contains a copy of it - everything that happens in happens somewhere in !

That said, it isn't true that homomorphic images of torsion-free groups are torsion-free. This is obvious if you know about free groups, but if you do not then take the integers under addition. It is an interesting exercise to prove that every homomorphic image of is finite.