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. Answer the same questions as in Exercise 21.

Data from Exercise 21

A student is asked to show that if H is a normal subgroup of an abelian group G, then G / H is abelian. The student's proof starts as follows:

We must show that G / H is abelian. Let a and b be two elements of G / H. 

a. Why does the instructor reading this proof expect to find nonsense from here on in the student's paper? 

b. What should the student have written? 

c. Complete the proof.