Free abelian group

Definition

Let $A$ be an abelian group, written additively. A basis of $A$ is a subset $B \subseteq A$ such that each $a \in A$ has exactly one expression $a = k_1 b_1 + \cdots + k_n b_n$ with distinct $b_i \in B$ and nonzero integers $k_i$. The group $A$ is a free abelian group when it has a basis.

Remarks

The additive group $\mathbb{Z}$ is free abelian with basis $\{1\}$, and $\mathbb{Z}^2$ is free abelian with basis $\{(1, 0), (0, 1)\}$. All bases of $A$ have the same cardinality, the rank, and a free abelian group of finite rank $n$ is isomorphic to $\mathbb{Z}^n$. The free abelian group on a set $S$ is the abelianisation of the free group $F_S$. Each subgroup of a free abelian group is free abelian.

Explanations

L1 human

A free abelian group is like a vector space over the integers with a basis. Each element is an integer combination of basis elements in exactly one way. The standard examples are $\mathbb{Z}^n$ for $n \geq 0$.