Representability of permutation representations on coalgebras and the isomorphism problem

Cristina Costoya, David Méndez, Antonio Viruel

Published: 20 Aug 2020, Last Modified: 29 Apr 2026Mediterranean Journal of MathematicsEveryoneRevisionsBibTeXCC BY-SA 4.0
Abstract: Let G be a group and let ρ: G → Sym(V ) be a permutation representation of G on a set V . We prove that there is a faithful G-coalgebra C such that G 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 through the G-action, and that the composition of the inclusion G ֒→ 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.