Prove that the torsion subgroup T of an abelian group G is a normal subgroup of G, and that G / T is torsion free. (See Exercise 22.) 

Data from Exercise 22

A torsion group is a group all of whose elements have finite order. A group is torsion free if the identity is the only element of finite order. A student is asked to prove that if G is a torsion group, then so is G / H for every normal subgroup H of G. The student writes, we must show that each element of G / H is of finite order. Let x ∈ G / H.