= Solution
A <group scheme> over $k$ is a $k$-scheme $G$ equipped with multiplication $m:G\times_kG\to G$, identity $e:\operatorname{Spec}k\to G$, and inversion $j:G\to G$ satisfying the associativity, identity, and inverse diagrams.
Consider
$$
q:G\times_kG\longrightarrow G,
\qquad q(x,y)=xy^{-1}.
$$
The <diagonal morphism> is $\Delta_{G/k}=q^{-1}(e)$. For a finite-type $k$-scheme the rational identity point is closed, so its inverse image is closed. Thus the diagonal is a closed immersion and every such group scheme over a field is a <separated scheme>.
In characteristic $p>0$, the infinitesimal additive group
$$
\alpha_p=\operatorname{Spec}k[t]/(t^p)
$$
is a nonreduced group scheme. Its comultiplication is $t\mapsto t\otimes1+1\otimes t$, which is well defined because $(t\otimes1+1\otimes t)^p=t^p\otimes1+1\otimes t^p$.
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