---
tag: 0BSCJA5G53
kind: example
name: Additive group of integers
aliases: [integers-under-addition]
msc: [20K99]
instance_of: [082GHNPJ53]
uses: [071DDFVG4J]
sources:
  - url: https://en.wikipedia.org/wiki/Integer
    locator: Algebraic properties
---

## 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|082GHNPJ53]].
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 $-1$.