Let $E$ be an elliptic curve over $\mathbb{F}_q$. I want to show $E(\mathbb{F}_q) \cong (\mathbb{Z}/m_1\mathbb{Z}) \times (\mathbb{Z}/m_2\mathbb{Z})$ where $m_1,m_2 \in \mathbb{Z}$ are such that
- $m_1 \mid m_2$;
- $m_1$ is the largest integer such that $F-1$ (with $F$ the $q$-th power Frobenius) is a multiple of $[m_1]$ in $\mathrm{End}(E)$.
I know numbers $m_1,m_2$ satisfying these two points exist as follows. We have some $m\in \mathbb{Z}$ such that $F-1 = \varphi \circ [m]$ for some $\varphi \in \mathrm{End}(E)$, for example $m=1, \varphi = F-1$, and the set of all such $m$ must be bounded, which can be seen from the degree of $F-1$.
Now take $m_1$ maximal amongst these $m$, and $\varphi \in \mathrm{End}(E)$ such that $F-1 = \varphi \circ [m_1]$. If we write $k$ for the number of elements in $E(\mathbb{F}_q)$ then we have that the degree of $F-1$ is $k$, for $F-1$ is separable and has kernel equal to $E(\mathbb{F}_q)$. So $\varphi \circ [m_1]$ is of degree $k = m_1^2 \cdot \deg(\varphi)$. Therefore, if we take $m_2 := m_1 \cdot \deg(\varphi)$, then $k= m_1 \cdot m_2$ and $m_1 \mid m_2$.
How can we now see that $E(\mathbb{F}_q) \cong (\mathbb{Z}/m_1\mathbb{Z}) \times (\mathbb{Z}/m_2\mathbb{Z})$ holds? I know $E(\mathbb{F}_q) = \ker(F-1) = \ker(\varphi \circ [m_1])$, but the problem seems to be we know little about $\varphi$.
If $E$ is an elliptic curve over $\mathbb{F}_q$, then the Weil pairing $E(\mathbb{F}_q)\times E(\mathbb{F}_q)\rightarrow \mathbb{F}_q^*$ shows that there exist positive integers $m_1,m_2$ such that $$ E(\mathbb{F}_q)\cong \mathbb{Z}/m_1 \mathbb{Z} \times \mathbb{Z}/m_2 \mathbb{Z}, $$ with $m_1\mid gcd(m_2,q-1)$, see Chapter III, Corollary $8.1.1$ in Silverman's book "The Arithmetic of Elliptic Curves". So the subjectivity of the Weil pairing should help.