# Real Numbers under Addition form Infinite Abelian Group

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## Theorem

Let $\R$ be the set of real numbers.

The structure $\struct {\R, +}$ is an infinite abelian group.

## Proof

From Real Numbers under Addition form Group, $\struct {\R, +}$ is a group.

Then:

and:

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.5$. Examples of groups: Example $81$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.1$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: The Group Property - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(1)$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(i)}$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.4$

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- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups