Exercise 3

Let $G$ be an abelian group.

1. Prove that $G^{n}=\{x^{n}\mid x\in G\}$, where $n$ is a positive integer, is a subgroup of $G$.

2. Suppose also that $G$ is finite and $p$ is a prime that divides $|G|$. Using the map $\phi:G\to G^{p}$ where $\phi(x)=x^{p}$, prove that $G^{p}$ is a proper subgroup of $G$.(Hint: consider Cauchy’s theorem (1.31).)