Abelian group

An abelian group is a set $$A$$ equipped with a binary operation $$+$$, known as addition, such that:


 * Addition is associative and commutative;
 * There is an element $$0 \in A$$, known as the additive identity (or just zero) for which $$a + 0 = a$$ for all $$a \in A$$; and
 * For each element $$a \in A$$ there is an element $$-a \in A$$, known as the additive inverse of $$a$$, for which $$a + (-a) = 0$$.

This is equivalent to saying that an abelian group is a group whose group operation is commutative.