Let H be an arbitrary group and let \(\rho :H\rightarrow {\text {Sym}}(V)\) be any permutation representation of H on a set V. We prove that there is a faithful H-coalgebra C such that H arises as the image of the restriction of \({\text {Aut}}(C)\) to G(C), the set of grouplike elements of C. Furthermore, we show that V can be regarded as a subset of G(C) invariant under the H-action and that the composition of the inclusion \(H\hookrightarrow {\text {Aut}}(C)\) with the restriction \({\text {Aut}}(C)\rightarrow {\text {Sym}}(V)\) is precisely \(\rho \). We use these results to prove that isomorphism classes of certain families of groups can be distinguished through the coalgebras on which they act faithfully.

中文翻译：

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代数上置换表示的可表示性和同构问题。

令H为任意组，令\（\ rho：H \ rightarrow {\ text {Sym}}（V）\）为H在集合V上的任何置换表示。我们证明了有一个忠实ħ -coalgebra Ç使得ħ产生作为限制的图像\（{\文本{AUT}}（C）\）到G ^（ç），该组的类群元素Ç。此外，我们表明，在H作用下，V可以看作是G（C）不变量的子集，并且包含物的组成\（H \ hookrightarrow {\ text {Aut}}（C）\）带有\（{\ text {Aut}}（C）\ rightarrow {\ text {Sym}}（V）\）的限制就是\（ \ rho \）。我们用这些结果证明，某些族群的同构类可以通过忠实地作用于它们的结合代数来区分。