In 14.4 of "Introduction to Affine Group Schemes" it is proved (!) that if $A$ represents a finite connected group scheme over a perfect field $k$ of characteristic $p$ then $A$ has the form $k[X_{1}, X_{2}, ..., X_{n}] / (X_{1}^{p^{e_{1}}}, ...., X_{n}^{p^{e_{n}}})$. But what about $\mu_{p} = k[X]/(X^{p}1)$? It is connected but not isomorphic to $k[X]/(X^{p})$ as $k$groups. They are isomorphic as $k$schemes. Does this theorem mean " ...... $A$ has the form $k[X_{1}, X_{2}, ..., X_{n}] / (X_{1}^{p^{e_{1}}}, ...., X_{n}^{p^{e_{n}}})$ up to isomorphism of $k$schemes"?
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Others can do this much better than I, but here's what's happening: to describe a group scheme of any kind, you need to talk about not only the underlying space, but also the law of composition on the group. In this case, the kernel of $[p]$ in the muliplicative group, you describe the law of composition by writing down the the comultiplication on the affine ring $k[X]/(X^p)$. This is simply $X\mapsto 1 \otimes X + X \otimes 1 + X \otimes X$. 

