On the set of grouplikes of a coring

We focus our attention to the set Gr(ℭ) of grouplike elements of a coring ℭ over a ring A. We do some observations on the actions of the groups U(A) and Aut(ℭ) of units of A and of automorphisms of corings of ℭ, respectively, on Gr(ℭ), and on the subset Gal(ℭ) of all Galois grouplike elements. Among them, we give conditions on ℭ under which Gal(ℭ) is a group, in such a way that there is an exact sequence of groups {1} → U(A ^g ) → U(A) → Gal(ℭ) → {1}, where A ^g is the subalgebra of coinvariants for some g ∈ Gal(ℭ).