Additive group of integers

Example

Let $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$ be the set of integers, with the usual addition $+$. Then $(\mathbb{Z}, +)$ is an abelian group. Its identity element is $0$, and the inverse of $n$ is $-n$.

Remarks

The group $(\mathbb{Z}, +)$ is infinite, and each element $n$ is the $n$-fold sum of $1$ or of $-1$. Thus the element $1$ generates the group. The integers with multiplication are not a group, because $2$ has no multiplicative inverse in $\mathbb{Z}$.

Explanations

L1 human

The integers with addition are the first example of a group for most students. Adding is associative, $0$ changes nothing, and $-n$ undoes $n$. Each element of the group is a sum of copies of $1$ or of $-1$.