**Analogy With Groups**

Groups can be axiomatized by the same diagrams (equivalently, operations) as a Hopf algebra, where *G* is taken to be a set instead of a module. In this case:

- the field
*K*is replaced by the 1-point set - there is a natural counit (map to 1 point)
- there is a natural comultiplication (the diagonal map)
- the unit is the identity element of the group
- the multiplication is the multiplication in the group
- the antipode is the inverse

In this philosophy, a group can be thought of as a Hopf algebra over the "field with one element".

Read more about this topic: Hopf Algebra

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**group**the multiplication is the multiplication in the

**group**the antipode is the inverse In this ...

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