Finding the order of another element of a group If in a group $G$, $a^5 = e$ , $e$ is the identity element of $G$. If $a,b \in G$ and $aba^{-1}=b^2$ then find the order of b I have got no clue how to go forward with it, except that perhaps we can construct a cyclic group of order 5 with a and thus the order of $G$ will be $5k$ , but how to go forward from there, I have got no inkling. Soham

Here are some hints:

1. Since $a^5=e$, what integer $n$ will satisfy $(a^{-1})^n=e$
2. Now combine what you know about $a$ and $a^{-1}$ along with the equation $aba^{-1}=b^2$ to find an equation where the $a$s disappear.