Through a series of examples (all involving at most $3\times 3$ matrices, as with bigger groups it is clearly impossible to do a correct computation on the board), we illustrate some of the pathologies that actions of algebraic groups can present over a field of characteristic $p>0$. For instance, the action of a finite group (scheme) on a curve might not lift to an action on the normalization. Then we will present some of the ways to deal with these problems, such as introducing a notion of "equivariant normalization", or working with "restricted" Lie algebras and the Frobenius map.