# Definition:Additive Group of Integers

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## Definition

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

Thus integer addition is:

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

## Also see

- Results about
**the 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: $\S 32$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**group** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**group**

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.In particular: Compare with Integers under Addition form Abelian Group and Additive Group of Integers is Countably Infinite Abelian Group, depending on whether the source specifically names this object or merely states its properties (or both) and whether infinitude is mentioned.If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

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