Claim on structures of groups in elliptic curves

Question

An old article makes the following claim.

This doesn't get a proof, so i suppose it must be easy. Is it a general property of abelian varieties of genus one or something more simple?

Answer 1

I suppose your $k$ is a finite field of characteristic $p$.

The point is that for any $r > 0$, we have either $E[p^r] \simeq \mathbb Z/p^r\mathbb Z$ or $E[p^r]$ is trivial. Here $E[p^r]$ means the group of $p^r$-torsion points of $E$ in the algebraic closure of $k$. This is proved in Silverman's book.

In view of this, the $p$-Sylow of $E(k)$, being a subgroup of $E[p^r]$ for some $r$, must be cyclic. That's why we must have only one component $\mathbb Z/p^{h_p}\mathbb Z$.

For any other prime $l\neq p$, we know that $E[l^r] \simeq (\mathbb Z/l^r \mathbb Z)^2$, hence the $l$-Sylow can have one or two component(s), therefore isomorphic to $\mathbb Z/l^{a_l}\mathbb Z \times \mathbb Z/l^{h_l - a_l}\mathbb Z$ for some integer $a_l$.