# Clarification/Help needed: for $H\lhd G$, a normal subgroup of index $(G:H)=n$, prove that $\forall g \in G, g^n \in H$.

I've been looking at this question, but I'm not sure where to go with it. I'm all sorts of confused here, and I'd be grateful if someone could explain the question and how to answer it.

Let $G$ be a group, $H \lhd G$ a normal subgroup of index $(G:H) = n$. Prove that for every $g \in G$, $g^n \in H$.

• 1. Step: Solve this when $H=\{1\}$. 2. Step: Reduce the general case to this :). – Martin Brandenburg Feb 19 '14 at 18:43
• if you are interested in such lemmas, Regardles of $H$ being normal or not , you can say $g^k\in H$ for some pozitive $k$ s.t $k\leq n$ – mesel Feb 19 '14 at 21:24

Hint: Interpret the question in the quotient group $G/H$.