Products, coproducts, and free objects in groups and abelian groups

Last post we discussed products, coproducts, and free objects generally in category theory. This post we focus our attention on and .

It is trivial to prove that is a product in and . It is also easy to see that the direct sum of Abelian groups is a coproduct in .

These definitions are external direct products or external direct sums, but sometimes a group has the direct product or direct sum structure within itself, and we may call it an internal weak direct product or internal direct sum in that case.

In light of this theorem, we have the following definition: We shall now construct a group that is free on the set . Let , then is the trivial group. If , let be a set disjoint from such that . Choose a bijection and denote the image of by . Choose a singleton that is disjoint from . A word on is a sequence with such that for some , for all . The constant sequence is called the empty word and is denoted . A word on is said to be reduced iff

Every nonempty reduced word is of the form where , , . Hereafter, this word is denoted by . Two reduced words and are equal iff both are 1 or and for each . Consequently the map from into the set of all reduced words on given by is injective. Identify with its image and consider it to be a subset of . Define a binary operation on the set of reduced words on by juxtaposition and cancellations of adjacent terms.

Some facts about free groups: if , then the free group is nonabelian. Every element except has infinite order. Every subgroup of a free group is itself a free group on some set. A consequence is that any group is isomorphic to a quotient group . is the free group on and is the kernel of the epimorphism . is determined to isomorphism by and is determined by any subset that generates it as a subgroup of . If is a generator of , then under the epimorphism , . The equation in is called a relation on the generators . A given group may be completely described by specifying a set of generators of and a suitable set of relations on these generators. This description is not unique since there are many possible choices of both and . Conversely, suppose we are given a set and a set of reduced words on the elements of . Let be a free group on and the normal subgroup of generated by (intersection of all normal subgroups of containing ). Let and identify with its image in under the map . Then is a group generated by and all the relations are satisfied.

Finally, we define the free product, which is the coproduct in .

Given a family of groups , let . Let be a singleton disjoint from . A word on is any sequence such that and for some , for all . A word is reduced iff

Let be the set of all reduced words on . forms a group called the free product of . is the identity element and the product of two words is juxtaposition with any necessary cancellations and contractions.

given by , is a monomorphism of groups. Finally, we discuss free abelian groups, the free objects in . The proof is straightforward but tedious.

In the next post, we discuss finitely generated abelian groups and their structures.