How to show rational points of finite order on an elliptic cure are closed under addition

Question

I would like to show that rational points of finite order on an elliptic curve are closed under addition.

If $P_1$ and $P_2$ are rational (actually integral) points of finite order, say $nP_1= O$ and $mP_2=O$,

I would like to say:

$$O=nmP_1 +nmP_2 =nm(P_1+P_2)$$

My question is how do I know the rightmost equality holds. Thanks

Answer 1

The rational points of an elliptic curve are an abelian group, i.e. a $\mathbf Z$-module and the points of finite order its torsion subgroup. The last equality is part of the distributive laws for modules.