Question 124
Let $k$ and $n$ be positive integers, and let $G$ be a group of order $n$. Prove that the following two statements are equivalent:
(a) The numbers $k$ and $n$ are relatively prime.
(b) For every subgroup $H$ of $G$, the set $\{x: x \in G \text{ and } x^k \in H\}$ is contained in $H$.
(a) The numbers $k$ and $n$ are relatively prime.
(b) For every subgroup $H$ of $G$, the set $\{x: x \in G \text{ and } x^k \in H\}$ is contained in $H$.