An additive group is a
group of which the group operation is to be thought of as ''addition'' in some sense. It is usually
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
, and typically written using the symbol + for its binary operation.
This terminology is widely used with structures equipped with several operations for specifying the structure obtained by forgetting the other operations. Examples include the ''additive group''
of the
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
and of a
ring. This is particularly useful with rings and
fields to distinguish the additive underlying group from the
multiplicative group of the
invertible elements.
References
{{DEFAULTSORT:Additive group
Algebraic structures
Group theory