## Normal Subgroup Whose Order is Relatively Prime to Its Index

## Problem 621

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$.

Suppose that the order $n$ of $N$ is relatively prime to the index $|G:N|=m$.

**(a)** Prove that $N=\{a\in G \mid a^n=e\}$.

**(b)** Prove that $N=\{b^m \mid b\in G\}$.