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).