The order of the element $gN$ in $G/N$ is $n$, where $n$ is the smallest positive integer such that $g^n\in N$. Give an example to show that the order of $gN$ in $G/N$ may be strictly smaller than the order of $g$ in $G$.
I have determined that such an example will be a case where $|g|>n$, but I'm not sure where to go from here. I find that coming up with examples is the hardest part for me. If anyone has any tips on how to come up with examples in general, they would be greatly appreciated.
Consider $\mathbb{Z}_4$ (with addition) and set $g=1$. Obviously $|g|=4$. Now take $N=\{0, 2\}$. It is a subgroup of order $2$ and $2g\in N$.
More generally: let $G$ be a finite cyclic group of order $n$ and pick a generator $g\in G$, i.e. $|g|=n$. Now take any nontrivial subgroup $1\neq N\leq G$ (which is normal since $G$ is abelian). Then $|G/N|<|G|$ because $N$ is nontrivial. In particular $|gN|$ divides $|G/N|$ (by Lagrange's theorem) and hence is strictly smaller then $|g|$.