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

• Additive Group of Integers is Countably Infinite Abelian Group

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.

• 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: $\S 32$

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