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Excerpt from Waterhouse, 14.4 Structure of Finite Connected groups.

Thm. Let $A$ represent a finite connected group scheme over a perfect field of characteristic $p$. Then $A$ has the form

$k[X_1, ..., X_n]/(X_1^{p^{e_1}}, \ldots, X_n^{p^{e_n}})$.

Proof. Let $I_A$ be the augmentation ideal of $A$. By connectedness, $I_A$ is nilpotent. [...] We assume therefore that

the Hopf subalgebra $B = A^p$ is one of these truncated polynomial algebras, say with generators $x_i$ and relations

$x_i^{q_i} = 0$. Choose $y_i \in A$ with $y_i^p = x_i$, and choose also a set $\{z_j\}$ in $A$ maximal with resprect to

the requirements that $z_j^p = 0$ and that the $z_j$ be linearly independent in $I_A/I_A^2$. Let $C = k[Y_i, Z_j]/(Y_i^ {pq_i}, Z_j^p$ which maps in the obvious way to $A$. We claim that this map is an iso.

Embed $A^p$ in $C$ by $x_i \mapsto Y_i^p$. Then $C$ is a free $B$-module. By the main theorem, $A$ is also free over $A^p$. As in (14.2), then, it is enough to show that $C/I_BC \to A/I_BA$ is an iso.

I don't understand the last sentence.

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