**Hopf Algebra Structure**

We give *kG* the structure of a cocommutative Hopf algebra by defining the coproduct, counit, and antipode to be the linear extensions of the following maps defined on *G*:

The required Hopf algebra compatibility axioms are easily checked. Notice that, the set of group-like elements of *kG* (i.e. elements such that and ), is precisely *G*.

Read more about this topic: Group Hopf Algebra

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