  In this note the smooth (i.e. with open stabilizers) linear and {\sl
semilinear} representations of certain permutation groups (such as infinite
symmetric group or automorphism group of an infinite-dimensional vector space
over a finite field) are studied. Many results here are well-known to the
experts, at least in the case of {\sl linear representations} of symmetric
group. The presented results suggest, in particular, that an analogue of
Hilbert's Theorem 90 should hold: in the case of faithful action of the group
on the base field the irreducible smooth semilinear representations are
one-dimensional (and trivial in appropriate sense).
