Let $G$ be a group and $H$ a subgroup of $G$. Consider the set $G/H$ of left cosets modulo $H$. The group G operates on $G/H$ on the left by $(g,xH)\mapsto gxH$. Let $N$ be the normalizer of $H$. The group $N$ operates on $G/H$ on the right by $(xH,n)\mapsto xHn=xnH$. This operation induces on H the trivial operation and hence, on passing to the quotient, $N/H$ operates on $G/H$ on the right.
I am not sure what is meant here by
...This operation induces on H the trivial operation...
The operation referred to is $$N\rightarrow\mathfrak{S}_{G/H},\ (n\mapsto(xH\mapsto xnH));$$
Is the induced operation $H\rightarrow\mathfrak{S}_{G/H},\ (h\mapsto(xH\mapsto xhH=xH))$?
Thank you in advance.
It means that if $h\in H, nhH=nH$ since $hH=H$ since $H$ is a subgroup and $h\in H$. $hH\subset H$ since $h\in H$ and $H$ a subgroup, let $h\in H, h'=h(h^{-1}h')$ and $h^{-1}h'\in H$.