Find the order of an element of finite group Let $G$ be a finite group and $g,h\in G-\{1\}$ such that $g^{-1}hg=h^2$.
In addition $o(g)=5$ and $o(h)$ is an odd integer. Find $o(h)$.
I know from a previous exercise that if there exists a natural number $i$ such that $g^{-1}hg=h^i$ then for all $n\in \mathbb{N}$, $g^{-n}hg^n=h^{i^n}$.
I thought I could use this fact somehow, but so far no luck.
Please give me a hint.

$h=g^{-5}hg^5=g^{-4}h^2g=g^{-3}h^4g^3=g^{-2}h^8g^2=g^{-1}h^{16}g=g^{32}$.
So $g^{31}=e$. Since $31$ is prime the order is $31$.

I used $g^{-1}h^n g=(g^{-1}h g)^n$ which is straightforward, it is proven when we prove conjugate of product is product of conjugates, which is prove when we show conjugating by an element is an isomorphism.
