In this post, we’ll only focus on additive abelian groups. By *additive*, we mean the underlying group operation is denoted by +. The identity and inverse of *x* are denoted by 0 and –*x* respectively. Similarly, 2*x*+3*y* refers to *x*+*x*+*y*+*y*+*y*. Etc etc.

Let *G* be an abelian group and *S* a subset of *G*. Consider <*S*>: the subgroup generated by *S*. As we saw earlier, this comprises of all “words” in *S*, i.e. 0, *a*, *b*, *c*, *a*+*b*, 2*a*+*c*, *a*–*b*+*a*, …, where *a*, *b*, *c*, … are elements of *S*. But since *G* is now abelian, we can write each term in the form: , where each *m _{i}* is an integer and

*a*‘s are distinct terms belonging to

_{i}*S*. Thus:

.

**Important note**. There are always finitely many terms in the sum, although if *S* is infinite, there is no upper bound to the number of terms. If *S* is finite, clearly the number of terms is upper-bounded by #*S*.

Summary. IfGis abelian, then <S> is easily described as the set of all finiteZ-linear combinations of elements ofS.

## Free Abelian Groups

Just like the case of free groups, we now consider an arbitrary set *S* and consider the set of all *formal* **Z**-linear combinations of *S*. By this, we mean the set of all symbolic expressions of finite-length as follows:

with the understanding that the expression is unchanged if we add or remove terms of the form 0·*x* (*x* in *S*). Addition between elements of *F _{ab}*(

*S*) is given term-wise, e.g. the sum of

*a*+3

*b*-2

*c*and

*b*+2

*c*–

*d*is

*a*+4

*b*–

*d*. The resulting group

*F*(

_{ab}*S*) is called the

**free abelian group**generated by

*S*.

**Universal Property**.

The free abelian group satisfies a similar universal property: let *i* : *S* → *F _{ab}*(

*S*) be the map which takes

*x*to 1·

*x*. Then for any function

*f*:

*S*→

*G*to an

*abelian*group

*G*, there exists a unique group homomorphism

*g*:

*F*(

_{ab}*S*) →

*G*such that

*gi*=

*f*.

The map *g* itself is easy to describe: it takes the formal sum (where ) to the element .

## Direct Sums and Direct Products

Here is when things get a little more abstract. Consider a collection of abelian groups *H _{i}*, where

*i*runs through a (possibly infinite) index set

*I*. We’ll construct two groups: the direct sum and the direct product.

Definition. Thedirect productof theH‘s is the set:_{i}where the group operation is given by component-wise product, i.e. . The

direct sumis the subgroup:

In short, the direct sum only contains those elements whose components are all 0 except possibly at finitely many places. For example, if we have *I* = {0, 1, 2, … } and each *H _{i}* =

**Z**, then the direct product is precisely the set of all integer sequences; while the direct sum is the set of all integer sequences with only finitely many non-zero entries.

In particular, for any set S, the free abelian group generated by S is the direct sum of (possibly infinitely many) copies of

Z, indexed by S. In symbolic form,

*No doubt the reader will ask: what’s the point of the two different definitions?*

In order to explain that, we have to go into universal properties again. Basically, the two constructions satisfy two different types of universal properties. One can even say, after looking at the diagrams, that the two constructions are *dual* to each other.

## More Universal Properties

First look at the direct product . For every index *i*, there is a projection map which takes (*x _{i}*)

*to the*

_{i}*i*-th component

*x*. This gives us a whole bunch of maps

_{i}*p*. The gist of the matter is as follows.

_{i}

Universal Property 1. For any abelian group K, together with a collection of homomorphisms , there is a unique homomorphism such that for every i.

In diagram form, we have:

How does the map *f* work? It takes an to the element which is an element of . The fact that *f* is a homomorphism follows from the fact that all the *q _{i}*‘s are.

Now let’s look at the direct sum . For each index *i*, we get an injection which takes to the element (…, 0, 0, …, 0, *x _{i}*, 0, … ) in

*G’*, i.e. all components are 0 except the

*i*-th component, which takes the value of

*x*.

_{i}

Universal Property 2. For every abelian group K and a collection of group homomorphisms , there is a unique homomorphism such that for each i.

In diagram form, we now have:

Let’s define the map *g*. Each element of is a multi-component , where all but finitely many *x _{i}*‘s are zero. So we can let

*g*take (

*x*) to . This is a well-defined sum since there’re only finitely many non-zero terms (we don’t know how to sum infinitely many terms without some form of convergence).

_{i}

Summary.We can describe the universal properties as follows. For direct product:For direct sum:

*Exercises. * Here’re some exercises to enhance your concepts of direct sum & direct product.

- In the universal property diagram for direct product, replace with instead. How does the universal property fail?
- In the universal property diagram for direct sum, replace with instead. How does the universal property fail?
- Replace “abelian groups” and “homomorphisms” with “sets” and “functions” respectively. Explain what the corresponding direct sum and direct product are in this context, i.e. find appropriate constructions which satisfy the universal properties.
- (
**Hard**) Replace “abelian groups” with “groups”. Find the corresponding direct sum and direct product, i.e. find constructions which satisfy the universal properties.*Note: the direct product is easy, but direct sum is much harder*!

*Hints for the Exercises*. The following are ROT-13 encoded.

- Fhccbfr k vf na ryrzrag bs X fhpu gung abar bs gur dv’f gnxr k gb gur vqragvgl. Gura s vf abg jryy-qrsvarq ba k. Va bgure jbeqf, s znl abg rkvfg.
- Guvf vf nyernql rkcynvarq va gur grkg. Vs jr unir na ryrzrag bs gur qverpg cebqhpg, jubfr pbzcbaragf ner nyy aba-mreb, gura jr znl unir gb fhz npebff vasvavgryl znal aba-mreb ryrzragf. Fb t vf abg jryy-qrsvarq.
- Gur qverpg fhz vf gur qvfwbvag havba bs gur frgf. Gur qverpg cebqhpg vf gur frg gurbergvp cebqhpg.
- Gur qverpg cebqhpg vf vqragvpny gb gur pnfr sbe noryvna tebhcf. Gur qverpg fhz vf zhpu gevpxvre: vg pbzcevfrf bs “jbeqf” jubfr punenpgref ner ryrzragf bs vaqvivqhny tebhcf, jurer arvtuobhevat punenpgref orybat gb qvfgvapg tebhcf. Tebhc cebqhpg vf qrsvarq ol pbapngrangvat jbeqf gbtrgure, naq erzbivat erqhaqnapl vs gur ynfg punenpgre bs gur cerivbhf jbeq orybatf gb gur fnzr tebhc nf gur svefg punenpgre bs gur arkg jbeq.