Hw11 Problem 3

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Problem 3

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 in $G/H$.

Solution

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

$a$ and $b$ should be elements of $G$, not $G/H$.

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(b) What should the student have written?

We want to look at coset elements of $G/H$, so we should let $aH, bH \in G/H$.

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(c) Complete the proof.

Let $aH, bH \in G/H$. Now, $(aH)(bH)=(ab)H$. We know $G$ is abelian, thus $ab=ba$ and it follows that $(ab)H=(ba)H=(bH)(aH)$. Therefore, $(aH)(bH)=(bH)(aH)$ and $G/H$ is abelian.