In mathematics, the category) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. Group theory may be thought of as the study of this category.

Relation to other categories

There are two forgetful functors from groups to monoids from groups to sets has two adjoints is the right adjoint functor the left adjoint functor sending every monoid to the Grothendieck group is the free functor the free group.

Categorical properties

The monomorphisms are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are precisely the bijective homomorphisms.

is both complete and co-complete. The category-theoretical product is just the direct product of groups while the category-theoretical coproduct is the free product of groups. The zero objects are the trivial groups (consisting of just an identity element).

has a category-theoretic kernel (given by the ordinary kernel of algebra), and also a category-theoretic cokernel (given by the quotient group by the normal closure is the kernel of its cokernel.

Not additive and therefore not abelian

The category of abelian groups, is a full subcategory is an abelian category isn't even an additive category, because there is no natural way to define the "sum" of two group homomorphisms. A proof of this is as follows: The set of morphisms from the symmetric group of ten elements would be a ring, so this finite ring would have no zero divisors. A finite ring with no zero divisors is a field by Wedderburn's little theorem, but there is no field with ten elements because every finite field has for its order, the power of a prime.

Exact sequences

The notion of exact sequence, and some results from the theory of abelian categories, such as the nine lemma, the five lemma.

is a regular category.

References