It is quite a common misunderstanding that $xH=Hx$ means that $xh=hx$ for any $h\in H$. This is generally false.
The assertion $xH=Hx$ means that
- for every $h\in H$, there exists $h_1\in H$ with $xh=h_1x$
- for every $h\in H$, there exists $h_2\in H$ with $hx=xh_2$
Consider the group $G=S_4$ and its normal subgroup $H=A_4$ (the even permutations). Since $[S_4:A_4]=2$, the subgroup $A_4$ is normal. However, taking $x=(12)$ and $h=(123)$, we have
However, $(132)(12)=(23)$, so in this case $h_1=(132)\ne h$.
We can also find two elements in $A_4$ that don't commute:
Note: the convention about function composition is the standard functional one, that is, on the left (think to $\circ$ between two cycles).