## Group element orders.

Let $G$ be an abelian group. Suppose $\alpha, \beta \in G$ are of finite order with $|\alpha| = m = m'p^{v'}$ and $|\beta| = n = p^v$ where $p$ is prime, $v' < v$ and $\mathrm{gcd}(m,n) = p^{v'}$ (i.e. $p$ does not divide $m'$).

Prove that $|\alpha \beta| = \mathrm{lcm}(m,n)$.