Let K be a field and G be a group of its automorphisms endowed with the compact-open topology, cf. Sect. 1.1. If G is precompact then K is a generator of the category of smooth (i.e. with open stabilizers) K-semilinear representations of G, cf. Proposition 1.1. There are non-semisimple smooth semilinear representations of G over K if G is not precompact. In this note the smooth semilinear representations of the group SΨ\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\mathfrak {S}}\nolimits _{\Psi }$$\end{document} of all permutations of an infinite set Ψ\documentclass[12pt]{minimal}
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\begin{document}$$\Psi $$\end{document} are studied. Let k be a field and k(Ψ)\documentclass[12pt]{minimal}
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\begin{document}$$k(\Psi )$$\end{document} be the field freely generated over k by the set Ψ\documentclass[12pt]{minimal}
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\begin{document}$$\Psi $$\end{document} (endowed with the natural SΨ\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\mathfrak {S}}\nolimits _{\Psi }$$\end{document}-action). One of principal results describes the Gabriel spectrum of the category of smooth k(Ψ)\documentclass[12pt]{minimal}
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\begin{document}$$k(\Psi )$$\end{document}-semilinear representations of SΨ\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\mathfrak {S}}\nolimits _{\Psi }$$\end{document}. It is also shown, in particular, that (i) for any smooth SΨ\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\mathfrak {S}}\nolimits _{\Psi }$$\end{document}-field K any smooth finitely generated K-semilinear representation of SΨ\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\mathfrak {S}}\nolimits _{\Psi }$$\end{document} is noetherian, (ii) for any SΨ\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\mathfrak {S}}\nolimits _{\Psi }$$\end{document}-invariant subfield K in the field k(Ψ)\documentclass[12pt]{minimal}
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\begin{document}$$k(\Psi )$$\end{document}, the object k(Ψ)\documentclass[12pt]{minimal}
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\begin{document}$$k(\Psi )$$\end{document} is an injective cogenerator of the category of smooth K-semilinear representations of SΨ\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\mathfrak {S}}\nolimits _{\Psi }$$\end{document}, (iii) if K⊂k(Ψ)\documentclass[12pt]{minimal}
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\begin{document}$$K\subset k(\Psi )$$\end{document} is the subfield of rational homogeneous functions of degree 0 then there is a one-dimensional K-semilinear representation of SΨ\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\mathfrak {S}}\nolimits _{\Psi }$$\end{document}, whose integral tensor powers form a system of injective cogenerators of the category of smooth K-semilinear representations of SΨ\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\mathfrak {S}}\nolimits _{\Psi }$$\end{document}, (iv) if K⊂k(Ψ)\documentclass[12pt]{minimal}
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\begin{document}$$K\subset k(\Psi )$$\end{document} is the subfield generated over k by x-y\documentclass[12pt]{minimal}
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\begin{document}$$x-y$$\end{document} for all x,y∈Ψ\documentclass[12pt]{minimal}
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\begin{document}$$x,y\in \Psi $$\end{document} then there is a unique isomorphism class of indecomposable smooth K-semilinear representations of SΨ\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\mathfrak {S}}\nolimits _{\Psi }$$\end{document} of each given finite length. Appendix collects some results on smooth linear representations of symmetric groups and of the automorphism group of an infinite-dimensional vector space over a finite field.
