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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

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    $nmP_1$ actually means $P_1 + P_1 + \cdots P_1$, and addition on the curve is commutative, so... – rogerl Feb 28 '15 at 21:42
  • @rogerl Thanks, that's what I thought and how I have the leftmost equality. My question is regarding the validity of saying this equals nm times this third point (P_1+P_2) where addition is on the curve. Maybe I am making too much of this, but by definition we know a point of finite order is as you say. I am stuck on the right side.Thanks –  Feb 28 '15 at 21:53

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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.

Bernard
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