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Let $E$ be an elliptic curve over a finite field $\mathbb{F}_q$ where $q$ is prime. Let $P$ be a point on $E$.

Consider the point $Q=(q+1)P=P+\cdots+P$, which is $P$ added to itself $q+1$ times. Due to the fact that we are in the finite field $\mathbb{F}_q$, I would expect the "scalar", in this case, $q+1$, to respect the ground field property and have $Q=P$, but this is not the case.

I am told that $E(\mathbb{F}_q)$ is either cyclic or a product of two cyclic groups, and that the order may not be $q$. Are there underlying concepts that I may be missing?

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    $\begingroup$ The basic point to make is that the addition operation on points on $E$ is not at all the same as adding the coordinates of the points, so you shouldn't expect it to obey the same rule. (You can find on Wikipedia the (somewhat complicated) formulas for the coordinates of the point $P+Q$ in terms of the coordinates of $P$ and of $Q$.) $\endgroup$ – user64687 Oct 30 '13 at 12:13
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You are right; you are missing some points. As of now you seem to be confusing the torsion points of elliptic curves with the (torsion elements of) multiplicative group $\mathbb F_q^*$. This is not a bad analogy at a higher level; but it is not superficially true. You will need to spend quite some time to understand everything. You can find a rigorous proof of this fact in Silverman's book on elliptic curves. If you want to understand heuristically, here is the following.

An elliptic curve over $\mathbb C$ is isomorphic to $S^1 \oplus S^1$ as a topological group, where $S^1$ is the circle group. The $m$-torsion has size $\mathbb Z/m\mathbb Z\oplus\mathbb Z/m\mathbb Z$. Due to the fact that elliptic curves are actually algebraic, and in particular, the torsion points are algebraic, these facts largely carry over to elliptic curves over algebraically closed char-$p$ fields(Here is where I am fudging things). The $\mathbb F_q$-rational points are obviously finite in number and they belong to some $m$-torsion group for the elliptic curve considered over the algebraic closure of $\mathbb F_q$. That it is either cyclic or a direct sum of two cyclic group now follows from the structure theorem of finitely generated abelian groups.

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