Let $(G,\circ)$ be a group and $N\subseteq G$ a normal subgroup of order $n<\infty$ and let $g\in G$. Is the element $g^n$ in $N$?
Given a subgroup $H\subseteq G$ of order $n$, is element $g^n$ in $H$ for all $g\in G$?
Edit: $g^n=\underbrace{g\circ g\circ\ldots\circ g}_{n\text{ times}}$