Let $G$ be a group, let $d \geq 2$ be an integer, and define a map $\phi(g)=g^d$, Prove that $PrePer(\phi, G)= G_{tors}$, i.e., prove that the preperiodic points are exactly the points of finite order in $G$
I have some problems to show that every point in $G_{tors}$ are in $PrePer(\phi, G)$, my try is for $g \in G_{tors}$ there is some $m$ such that $g^m = e$. My guest we can find $k$ such that $d^k \equiv 1 (m)$, if that happened Im ready, but im not sure how to prove that. Any help please.
Let $m$ be the order of $g$. Now consider the sequence $d,d^2,d^3,\ldots$ modulo $m$. Since $\mathbb Z/m\mathbb Z$ is finite, there are $j,k$ such that $1\leq j\leq k$ and $d^j\equiv d^k\bmod m$. Hence, $g^{d^k-d^j}=e$, implying immediately that $g^{d^k}=g^{d^j}$, i.e. $g$ is pre-periodic.