Frobenius kernels of algebraic supergroups and Steinberg's tensor product theorem

For a split quasireductive supergroup $G$ defined over a field, we study structure and representation of Frobenius kernels $G_r$ of $G$ and we give a necessary and sufficient condition for $G_r$ to be unimodular in terms of the root system of $G$. We also establish Steinberg’s tensor product theorem for $G$ under some natural assumptions.