Free abelian group

Definition

Expressive definition with explicit generating set

This is not the rigorous definition, but is the one that is most useful for explicit expressions and computations.

Suppose $A$ is a set (with no additional structure necessary). The free abelian group on $A$ (or the free abelian group with generating set $A$) is a group whose elements are defined as formal finite $\mathbb{Z}$-linear combinations of elements of $A$, i.e., finite sums of the form:

$m={\sum }_{a\in A}{m}_{a}a$

where $m_a\in \mathbb{Z}$ for all $a$ and $m_a\ne 0$ for only finitely many $a\in A$. The addition rule is as follows: we group together coefficients of the same element of $A$. Thus, if $m={\sum }_{a\in A}{m}_{a}a$ and $n={\sum }_{a\in A}{n}_{a}a$, then $m+n={\sum }_{a\in A}(m_a+n_a)a$.

When writing a formal sum, we ignore all the terms with zero coefficient, and use subtraction to denote the addition of something with a negative coefficient (so $a+(-1)b$ is written as $a-b$).

Note that if $A$ already had an additive structure, that structure is ignored for our purposes. A more precise formulation of this would be to not use the elements of $A$ themselves but a generating set equipped with a bijection to $A$; however, this extra layer of formality is often unnecessary.

Definition in terms of rank

Let $\alpha$ be a cardinal (a nonnegative integer if finite, otherwise an infinite cardinal). The free abelian group of rank $\alpha$ is defined as the free abelian group on any set of size $\alpha$. This is unique up to isomorphism: any bijection between two sets induces an isomorphism of the corresponding free abelian groups. In fact, something stronger is true: free abelian groups arising from sets of different cardinalities are not isomorphic, so the rank of a free abelian group is a unique cardinal.

Examples

Group	Freely generating set for which it can be viewed as a free abelian group	Proof/explanation
group $C_n(X)$ of singular n-chains in the singular chain complex of a topological space $X$	set $S_n(X)$ of singular n-simplices	by definition
group $C_0(X)$ of singular 0-chains of a topological space $X$	underlying set of $X$	by definition plus the observation that $S_0(X)$ is canonically identified with $X$.
zeroth homology group	set of path components	zeroth homology is free on set of path components