Model checking in finite fields and finite groups

We prove the following results. 1, First order model checking is fixed-parameter tractable on the class of finite fields, as a corollary of results of Ax on the theory of (pseudo)finite fields. 2. Every hereditary graph class first order definable in the class of finite groups is monadically stable, and thus has fixed-parameter tractable first order model checking. 3. Monadic second order model checking is not slicewise polynomial on the class of cyclic groups of prime-power order, assuming E $\neq$ NE. Thus the same is true on the class of finite fields. 4. The class of finite fields is finitely axiomatizable in monadic second order logic, and so there are no pseudofinite fields in this setting.