On the structure of $p$-schemes

We introduce and study an analog of $p$-groups in general scheme theory. It is proved that a scheme is a $p$-scheme if and only if so is each homogeneous component of it. Moreover, the automorphism group of a $p$-scheme is a $p$-group, and the $2$-orbit scheme of a permutation group $G$ is a $p$-scheme if and only if $G$ is a $p$-group. Both of these statements follow from the fact that the class of $p$-schemes is closed with respect to extensions.