Let $(C,\Delta,\varepsilon)$ be a coalgebra over a field $k$ .

Definition. The element $g\in C$ is called grouplike iff $g\neq 0$ and $\Delta(g)=g\otimes g$ . The set of all grouplike elements in a coalgebra $C$ is denoted by $G(C)$ .

Properties. $0)$ The set $G(C)$ can be empty, but (for example) if $C$ can be turned into a bialgebra, then $G(C)\neq\emptyset$ . In particular Hopf algebras always have grouplike elements.

$1)$ If $g\in G(C)$ , then it follows from the counit property that $\varepsilon(g)=1$ .

$2)$ It can be shown that the set $G(C)$ is linearly independent.