On the set of grouplikes of a coring

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