Students often write nonsense when first proving theorems about factor groups. The next two exercises are designed to call attention to one basic type of error, 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.
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