How to prove that (r,*) is an abelian group.

You need to firstly show that is a group, which requires that is closed under , that is associative,, that has an identity element, and that every element in has a corresponding inverse in . Then for to be abelian, you also need to show that is commutative.

But suppose and , where . For instance, take and . Then



which means is not closed under and is therefore not a group, and certainly not an abelian one.