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Let $G$ be a group and let $\rho\colon G\to\operatorname{Sym}(V)$ be apermutation representation of $G$ on a set $V$. We prove that there is afaithful $G$-coalgebra $C$ such that $G$ arises as the image of the restrictionof $\operatorname{Aut}(C)$ to $G(C)$, the set of grouplike elements of $C$.Furthermore, we show that $V$ can be regarded as a subset of $G(C)$ invariantthrough the $G$-action, and that the composition of the inclusion$G\hookrightarrow\operatorname{Aut}(C)$ with the restriction$\operatorname{Aut}(C)\to\operatorname{Sym}(V)$ is precisely $\rho$. We usethese results to prove that isomorphism classes of certain families of groupscan be distinguished through the coalgebras on which they act faithfully.

Representability of permutation representations on coalgebras and the isomorphism problem