Sometimes I see in books the term "additive abelian groups". In my opinion, when we use addition to represent the group operation, we already have in mind that the operation is commutative. So additive group means abelian group. Am I wrong? Are there "additive non-abelian groups"?

I quote this from a book:

"...it is shown that any additive group $M$ admits a scalar multiplication by integers, and if $M$ is abelian, the properties are satisfied to make $M$ a $Z$-module ..."

Why the author needs to say "if $M$ is abelian", given that it is said to be additive?

If the addition is not assumed to be abelian, then it is a general binary operation, so the author was saying " ... it is shown that any group $M$ admits a scalar multiplication by integers and if $M$ is abelian, the preperties are satisfied to make $M$ a $Z$-module ..." Right?

• The term "additive group" is really just kind of instructive - it encourages you to think of the binary operation as addition in some way. So yes, in general these groups will be abelian. – Edward ffitch Jul 4 '14 at 22:59
• Per mathworld.wolfram.com/AdditiveGroup.html the additive notation is preferred for use with Abelian groups. – Vladhagen Jul 4 '14 at 22:59
• “Additive” refers to the notation used for the operation; it has nothing to do with abeliannes; it's a fact that usually (but not necessarily) addition is used for commutative operations. – egreg Jul 4 '14 at 22:59
• @user139981 There are some generalizations of rings, where addition is not necessarily commutative, but the set is a group with respect to the operation. It's just notation; using $+$ instead of $\circ$ for denoting the operation on $S_4$ wouldn't make the group abelian. – egreg Jul 4 '14 at 23:05
• The symbol $+$ is used by lots of language to denote concatenation of strings or lists, and that operation is quite non-abelian. – Mariano Suárez-Álvarez Jul 4 '14 at 23:19

In additive abelian group the word ‘additive’ refers to the symbol used for the operation $({+})$ and, in principle, it has nothing to do with the group being abelian. It's true that in most cases the additive notation is used for abelian groups (or, more generally, for commutative operations), but this is not universal.