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

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