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

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4 Answers 4

up vote 8 down vote accepted

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.

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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.

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An group is a set $G$ with an operation $+$ satisfying three axioms:

1.) Associativity: $(a+b)+c = a+(b+c)$

2.) Identity: There exists an element $e \in G$ satisfying $e + a = a + e = a$ for any $a \in G$. In an additive group, $e$ is denoted with the symbol $0$.

3.) Inverses: for every $a \in G$ there exists an inverse denoted $-a$ such that $a + -a = 0$.

Satisfying these three axioms makes $(G,+)$ a group, and the fact that the identity is called $0$, inverses are denoted with a $-$ and the operation looks like $+$ makes the group additive. As opposed to a multiplicative group where the operation looks like $\cdot$ or $\times$, inverses look like $\frac{1}{a}$ or $a^{-1}$ and the identity looks like $1$.

If any set satisfies these axioms, it is called an additive group. Having other properties doesn't make it less of a group. For example $(\mathbb{R},+)$ is a group, and the real numbers are additionally a field, and they have an ordering $\leq$. The complex numbers also form an additive group, and are a field, but have no ordering. Each of these properties is looked at independently.

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Additive groups are typically assumed abelian. –  Zhen Lin Jan 19 '13 at 20:44
    
That's but one particular axiomatization of groups. There are many others, with varying number of operations and axioms (not necessarily equational). In many contexts one wants to consider these various notions of groups as equivalent. This is done in universal algebra and category theory (e.g. Hall's clones, Lawevere's algebraic theories, monads, etc). –  Math Gems Jan 19 '13 at 21:18

I think the answer is given by wikipedia: Additive group.

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