Proof that equation is solvable in abelian group Let $G$ be an abelian group, $a \in G$ be an element of finite order, $(\text{ord} \, a, n) = 1$. Prove that the equation $x^n = a$ is solvable in the group $G$.
I tried to apply a corollary from Lagrange's theorem, but I am concerned that the group can be of infinite order, and I do not know what to do with it

let $m= \text{ord}(a)$. and pick $k$ such that $k$ is congruent to the inverse of $n\pmod m$. We thus can write $kn$ as $dm+1$.
Notice that $(a^k)^n = a^{kn}=a^{dm+1} = a^{dm}\cdot a = e\cdot a = a.$
