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

On semilinear representations of the infinite symmetric group