Q:

Is the category of finite group schemes closed under fibred product?

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?

A:

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.

