Theorem 11.9

Statement:
Let $(a_1,a_2,\dots,a_n)\in\prod^n_{i=1} G_i$.If $a_i$ is of finite order $r_i$ in $G_i$, then the order of $(a_1,a_2,\dots,a_n)$ in $\prod^n_{i=1} G_i$ is equal to the least common multiple of all the $r_i$.

Proof:

This follows by a repetition of the argument used in the proof of Theorem 11.5. For a power of $(a_1,a_2,\dots,a_n)$ to give $(e_1,e_2,\dots,e_n)$ the power must simultaneously be a multiple of $r_1$ so that this power of the first component $a_1$ will yield $e_1$, a multiple of $r_2$, so that this power of the second component $a_2$ will yield $e_2$, and so on.