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?