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?
Best Answer
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.
For instance, the two operations on near-rings are usually denoted by addition and multiplication, but addition is not required to be commutative although a near-ring must be a group with respect to addition (see http://en.wikipedia.org/wiki/Near-ring).