Intermediate Group Theory (6)

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, 2x+3y 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, 2a+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: $m_1 a_1 + m_2 a_2 + \ldots + m_n a_n$ , where each m_i is an integer and a_i‘s are distinct terms belonging to S. Thus:

$\left<S\right> = \{m_1 a_1 + m_2 a_2 + \ldots + m_n a_n : m_i\in \mathbf{Z}, a_i\in S\}$ .

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. If G is abelian, then <S> is easily described as the set of all finite Z-linear combinations of elements of S.

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:

$F_{ab}(S)=\{ m_1 a_1 + m_2 a_2 + \ldots + m_n a_n : m_i\in\mathbf{Z}, a_i\in S\},$

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+3b-2c and b+2c–d is a+4b–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 $m_1 a_1 + \ldots + m_n a_n$ (where $m_i\in \mathbf{Z}, a_i\in S$ ) to the element $m_1 f(a_1) + \ldots + m_n f(a_n)$ .

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. The direct product of the H_i‘s is the set:

$G = \prod_{i\in I} H_i = \{(x_i)_{i\in I} :x_i\in H_i\}$

where the group operation is given by component-wise product, i.e. $(x_i) + (y_i) = (x_i + y_i) \in \prod H_i$ . The direct sum is the subgroup:

$G' = \oplus_{i\in I} H_i =\{(x_i) \in \prod H_i : x_i \ne 0 \text{ for only finitely many } i\}.$

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 $\prod H_i$ is precisely the set of all integer sequences; while the direct sum $\oplus H_i$ 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,

$F_{ab}(S) = \oplus_{i\in S} \mathbf{Z}.$

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 $G=\prod_i H_i$ . For every index i, there is a projection map $p_i : G \to H_i$ which takes (x_i)_i to the i-th component x_i. This gives us a whole bunch of maps p_i. The gist of the matter is as follows.

Universal Property 1. For any abelian group K, together with a collection of homomorphisms $q_i : K\to H_i$ , there is a unique homomorphism $f:K\to G=\prod_i H_i$ such that $p_i\circ f=q_i$ for every i.

In diagram form, we have:

How does the map f work? It takes an $x\in K$ to the element $(q_i(x))_{i\in I}$ which is an element of $\prod_i H_i$ . 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 $G' = \oplus_i H_i$ . For each index i, we get an injection $e_i : H_i \to G'$ which takes $x_i \in H_i$ 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 $f_i:H_i \to K$ , there is a unique homomorphism $g:\oplus_i H_i \to K$ such that $g\circ e_i = f_i$ for each i.

In diagram form, we now have:

Let’s define the map g. Each element of $\oplus_i H_i$ is a multi-component $(x_i)\in \prod_i H_i$ , where all but finitely many x_i‘s are zero. So we can let g take (x_i) to $\sum_i f_i(x_i)$ . 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).

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

$\prod_{i\in I} \text{Hom}(K, H_i) \cong\text{Hom}(K, \prod_{i\in I} H_i).$

For direct sum:

$\prod_{i\in I} \text{Hom}(H_i, K) \cong\text{Hom}(\oplus_{i\in I}H_i, K).$

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

In the universal property diagram for direct product, replace $\prod_i H_i$ with $\oplus_i H_i$ instead. How does the universal property fail?
In the universal property diagram for direct sum, replace $\oplus_i H_i$ with $\prod_i H_i$ 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.

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