Element in compact connected Lie group

Question

Well, I just saw a magical result that said given an element $g$ in a compact connected Lie group $G$ and a natural number $n$, then there always exist another element in $G$ such that $g=h^n$. Well, I really don’t see how can this be true, even don’t know which aspects of the theory of Lie group can bring this result. Any suggestions?

Answer 1

For compact connected Lie groups, the exponential map $\exp : \mathfrak{g}\to G$ is surjective.

Let $g\in G$ and $X\in \mathfrak{g}$ such that $g=\exp(X)$.

For $n\in\mathbb{N}^*$, just take $h:=\exp(\frac{X}{n})$.

Clearly, $h^n=g$.