Is the category of finite group schemes closed under fibred product? If not, what is the simplest counterexample? More precisely
Let $G_1$ and $G_2$ be two finite group schemes over $k$. Let $f_1$ and $f_2$ be two homomorphisms of group schemes from $G_1$ and $G_2$ to $G$ respectively, where $G$ is another finite group scheme. Is ir true that $G_1 \times_G G_2$ is also finite group scheme?
Yes, the fiber product $G_1\times_G G_2$ is a group scheme in a canonical way (finiteness is automatic since it is preserved by fiber products). This is easiest to see from the functor of points perspective. For any $k$-scheme $S$, the set of $S$-points of $G_1\times_G G_2$ is just the fiber product of sets $G_1(S)\times_{G(S)}G_2(S)$. But the fiber product of a pair of homomorphisms of groups has a natural group structure: $G_1(S)\times_{G(S)} G_2(S)$ is just the subgroup of the direct product $G_1(S)\times G_2(S)$ consisting of elements whose coordinates have the same image in $G(S)$ (this is a subgroup since the maps $G_1(S)\to G(S)$ and $G_2(S)\to G(S)$ are homomorphisms). So $G_1(S)\times_{G(S)}G_2(S)$ has a groups structure which is natural in $S$, which makes $G_1\times_G G_2$ a group scheme.