Yes it is true. But not for the commutative hopf algebras in general. You need some more assumptions: first we need the field to be algebraically closed and of characteristic zero. You also need cocommutativity and finite dimensionality of the hopf algebra to have a full "correspondence".To be more precise:
it can be shown that there is an equivalence of Categories, between the Category of commutative, cocommutative, finite dimensional Hopf algebras (over an algebraically closed field, of characteristic zero) and the category of the finite, abelian groups. It is possible to construct fully faithful and essentially faithful functors between and .
Start from an object of i.e. a commutative, cocommutative, finite dimensional Hopf algebra , over an algebraically closed field, of characteristic zero. The set of its grouplike elements forms a finite abelian group i.e. an element of . It is relatively easy to see that a hopf algebra morphism induces an abelian group homomorphism.So you get a functor
On the other hand, start from a finite abelian group and take its group hopf algebra . It is clearly commutative, cocommutative and finite dimensional, i.e. an object of . On the other hand, an abelian group homomorphism induces -by linear extension, due to the universal property of the group algebra- a morphism of hopf algebras between the corresponding group hopf algebras.So you get a functor
Now, it can be shown that:
Consequently, the functors , constitute an equivalence of the categories , .
In my understanding, it is actually this equivalence of categories, which inspired the introduction of the term quantum groups (implying that the hopf algebra theory may be considered as a kind of "quantum" generalization of the group theory).
Not exactly what you’re looking for?