Representability of Permutation Representations on Coalgebras and the Isomorphism Problem

Let H be an arbitrary group and let ρ :H→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 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↪Aut(C) with the restriction Aut(C)→Sym(V) is precisely ρ . 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.