Category of groups

In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.

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

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

The category of abelian groups, Ab, is a full subcategory of Grp. Ab is an abelian category, but Grp is not. Indeed, Grp isn't even an additive category, because there is no natural way to define the "sum" of two group homomorphisms. The identity function is an automorphism of every group, but the natural sum of this automorphism with itself would be a function which takes every element to its square. This is an endomorphism of a group if and only if it is abelian: $(xy)^2\; =\; x^2y^2$ if and only if $xyxy\; =\; xxyy$, if and only if $yx\; =\; xy$, in other words, when any two elements commute.

More precisely, there is no way to define the sum of two group homomorphisms to make Grp into an additive category. The set of morphisms from the symmetric group of order three to itself, $E=operatorname\{Hom\}(S\_3,S\_3)$, has ten elements: an element "z" whose product on either side with every element of "E" is "z" (the homomorphism sending every element to the identity), three elements such that their product on one fixed side is always themself (the projections onto the three subgroups of order two), and six automorphisms. If Grp were an additive category, then this set "E" of ten elements would be a ring. In any ring, the zero element is singled out by the property that 0"x"="x"0=0 for all "x" in the ring, and so "z" would have to be the zero of "E". However, there are no two nonzero elements of "E" whose product is "z", so this finite ring would have no zero divisors. A finite ring with no zero divisors is a field, but there is no field with ten elements. Since no such field exists, "E" is not a ring and Grp is not an additive category.

Every morphism "f" : "G" → "H" in Grp has a category-theoretical kernel (given by the ordinary kernel of algebra ker f = {"x" in "G" | "f"("x") = "e"}), and also a category-theoretical cokernel (given by the factor group of "H" by the normal closure of "f"("H") in "H"). Unlike in abelian categories, it is not true that every monomorphism in Grp is the kernel of its cokernel.

The notion of exact sequence is meaningful in Grp, and some results from the theory of abelian categories, such as the nine lemma, the five lemma, and their consequences hold true in Grp. The snake lemma however is not true in Grp.