Question

Prove that H c G is a normal subgroup if and only if every left coset is a right coset, i.e., aH = Ha for all a e G

Answer 1

$\Rightarrow$

Suppose first that $H\subset G$ is a normal subgroup. Then by definition we must have for all $a\in H$, $xax^{-1} \in H$ for every $x\in G$. Let $a\in G$ and choose $(ab)\in aH$ ($b\in H$). By hypothesis we have $aba^{-1} =abbb^{-1}a^{-1}=(ab)b(ab)^{-1} \in H$, i.e. $aba^{-1}=c$ for some $c\in H$, thus $ab=ca \in Ha$. So we have $aH\subset Ha$. You can prove $Ha\subset aH$ in the same way.

$\Leftarrow$

Suppose $aH=Ha$ for all $a\in G$. Let $h\in H$, we have to prove  $aha^{-1} \in H$ for every $a\in G$. So, let $a\in G$. We have that $ha^{-1} =a^{-1}h'$ for some $h'\in H$ (by the hypothesis). hence we have $aha^{-1}=h' \in H$. Because $a$ was chosen arbitrarily  we have the desired .