I need some help in relation to this exercise
"Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$."
I'm not allowed to use quotient groups in the proof, because the exercise is in the chapter before.
I tried by induction on $n$. The case $n=1,n=2$ are obvious, but even the case $n=3$ is giving me trouble so I give up studying the general case of the inductive step.
My other approach was studying left or right coset of $G$. But I only proved that if $g \in aH$ then $g^2 \notin aH$ if $a \notin H$, and I can't find a way to demonstrate that $g^n \in H$. (my starting idea was to prove that every power of $g$ is in a different coset but then I realize that in this way I don't handle several case, for example $g$ has period strictly lesser than $n$ and in conclusion it doesn't prove the exercise) Maybe I'm missing something about indexes, and this is why I asked here for some help,
(I can't use quotient groups because they are introduced later than this exercise, forgot to add this info in the beginning) Thanks in advance :)