Representability of Permutation Representations on Coalgebras and the Isomorphism Problem

Abstract

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.