Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$.

Suppose that $p$ is split in $K$ so that $E$ is ordinary, and write $p\cal{O}_K=\frak{p}\overline{\frak{p}}$. I would like to consider two subgroup schemes $E[\frak{p}]$ and $E[\overline{\frak{p}}]$ of $E[p]$ and ask the following two questions:

- Is it true in general that $\text{rank}\ E[\frak{p}]$= $\text{rank}\ E[\overline{\frak{p}}]=p$ ? I think I have an argument that goes through lifting to characteristic zero and the Main Theorem of CM but probably there is a simpler way.

Secondly,

- Is it true that $E[\frak{p}]$ and $E[\overline{\frak{p}}]$ can be seen as connected and étale parts (or viceversa) of $E[p]$? In the setting I have chosen for this question I don't see why this should be true (for example what distinguish $\frak{p}$ from $\overline{\frak{p}} $ ?) but it seems reasonable when I see $E/k$ as the reduction at the appropriate prime (above $\frak{p}$ or $\overline{\frak{p}} $) of a CM elliptic curve over a number field.

Thank you for your help.