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.

try finishing off this line of argument...
$$
\begin{align}
h &= ehe \\&= g^{-5}hg^5 \\&= g^{-4} (g^{-1}h g) g^4 \\&= g^{-4}h^2 g^4 
\\&=g^{-3}(g^{-1}h^2g)g^3 \\&=g^{-3}h^4g^3 \\&= \dots \\&=h^?
\end{align}
$$
so $h^{?-1}=e$
