What is an additive group?

Is an additive group a group which only has an addition operation, or can it also have other operations on it?

Thanks

Solution 1:

Let $G$ be some set. By calling $G$ an additive group, one typically wants to say that

• $G$ is an abelian group with respect to the operation in question,
• the group operation is denoted by "$+$",
• the identity element is denoted by "$0$",
• the inverse of an element $g$ is denoted by "$-g$",

so the term "additive group" actually refers to a set of conventions regarding notation. The only mathematically meaningful property is that $G$ is required to be abelian.

This denotation is useful when dealing with concepts where multiple group structures are involved (like fields) or when saying "additive group" for some other reason makes immediately evident which group one is refering to (like the additive group of a ring or a vector space).

Of course, as others have already pointed out, there may very well be additional group structures declared on an additive group as ultimately you're just looking for additional group structures on sets which already carry the structure of an abelian group.

Solution 2:

There certainly can be extra operations, but they play no part in it being an additive group. For example $\mathbb{Q}$ is an additive group under $+$; the fact that there is also a multiplication defined on it has no influence on it being an additive group.