Assume that $G$ is a group. Also, $a$ is a torsion element of $G$ such that $a^m=1$, for some natural number $m$. Also, there exists an element $y \in G $ such that $yay^{-1}=a^k \neq a$, when $gcd(k,m)=1$ and $1<k<m$. Can we conclude that $\langle a,y\rangle $ is a finite group?
Notice that $\langle a\rangle$ is a normal subgroup of $\langle a,y\rangle$. In addition, $\langle a,y\rangle/\langle a\rangle \cong \langle y\rangle/(\langle a\rangle \cap \langle y\rangle )$. Also, we know that for any natural number $t$, $b^tab^{-t}=a^{k^t}$. We have $gcd(k,m)=1$, so for some natural number $t$, $b^tab^{-t}=a$.