For a Hopf algebra $H$ and element $h \in H$ is called grouplike is $\Delta(h) = h \otimes h$. The identity $1_H$ is clearly grouplike, but in general non-trivial grouplike elements may fail to exist. See this question for Is there a name for a Hopf algebra whose only grouplike element if the identity?

For a $k$-Hopf algebra $H$ and element $h \in H$ is called grouplike is $\Delta(h) = h \otimes h$ and $\epsilon(h)=1_k$ ($\epsilon$ is the counit). The identity $1_H$ is clearly grouplike, but in general non-trivial grouplike elements may fail to exist. See this question for Is there a name for a Hopf algebra whose only grouplike element if the identity?