Page "Group object" Paragraph 19

* The group objects in the category of groups ( or monoids ) are the Abelian groups.

The reason for this is that, if inv is assumed to be a homomorphism, then G must be abelian.

More precisely: if A is an abelian group and we denote by m the group multiplication of A, by e the inclusion of the identity element, and by inv the inversion operation on A, then ( A, m, e, inv ) is a group object in the category of groups ( or monoids ).

Conversely, if ( A, m, e, inv ) is a group object in one of those categories, then m necessarily coincides with the given operation on A, e is the inclusion of the given identity element on A, inv is the inversion operation and A with the given operation is an abelian group.

See also Eckmann-Hilton argument.