# Definition:Additive Group of Integers

## Definition

The **additive group of integers** $\struct {\Z, +}$ is the set of integers under the operation of addition.

## Also see

Thus integer addition is:

- Well-defined on $\Z$
- Closed on $\Z$
- Associative on $\Z$
- Commutative on $\Z$
- The identity of $\left({\Z, +}\right)$ is $0$
- Each element of $\struct {\Z, +}$ has an inverse.

- Results about
**Additive Group of Integers**can be found here.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.5$. Examples of groups: Example $80$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Examples of Group Structure: $\S 32$

- 1992: William A. Adkins and Steven H. Weintraub:
*Algebra: An Approach via Module Theory*... (previous) ... (next): $\S 1.1$: Example $1$