Additive group of real numbers

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

• This group is the set of real numbers under addition.
• This group is the additive group of the field of real numbers.

This group is often denoted $(\mathbb {R} ,+)$ or simply as $\mathbb {R}$.

Group properties

Property	Satisfied	Explanation
Abelian group	Yes	Addition of real numbers is commutative
Nilpotent group	Yes	Abelian implies nilpotent
Simple group	No	Any non-trivial proper subgroup is a normal subgroup since abelian. It has such subgroups, for example additive group of rational numbers

See also

• Multiplicative group of non-zero real numbers
• Additive group of complex numbers
• Additive group of real algebraic numbers
• Additive group of rational numbers
• Additive group of integers