Group schemes appearing in the $p$-torsion of an abelian scheme

Question

This question is motivated by this one. In that question, there is stated without reference the fact that for an abelian variety over a field of characteristic $p$ (which I assume means the base field is algebraically closed), the only group schemes which appear as the $p$-torsion points are products of $\mathbb{Z}/p\mathbb{Z}$, $\mu_p$, and $\alpha_p$. I have two questions:

1. Does anyone have a reference for this fact? (I do not believe it follows immediately from the fact that the only group schemes of order $p$ over $k = \bar{k}$ are the ones listed above)
2. What is known for abelian schemes? Is there a classification of such group schemes?

Answer 1

The statement in the posting you make reference to is false. The $p$-torsion of a supersingular elliptic curve is a group-scheme, call it $\Gamma$, that fits into a nonsplitting exact sequence $$ 0\longrightarrow\alpha_p\longrightarrow\Gamma\longrightarrow\alpha_p\longrightarrow0 \,.$$