Examples of non-abelian groups under addition

Question

I tried to find non-abelian groups under addition, but failed. It seems to me that all number sets hold addition under a commutative law. What are some examples of non-abelian groups under addition?

Answer 1

Usually when we use the word "addition" it is for an operation that is commutative. So almost by definition, you will have a hard time finding a non-abelian group whose operation is called "addition."

Answer 2

A group involves a set and a binary operation on them, satisfying some conditions. Most common examples are set of numbers with addition as the binary operation. This is commutative. And so any subgroup of numbers under addition will be commutative (abelian) too.

So what you are expecting is a group on some other set where the binary operation is NOT commutative. There are so many sets with so many operations and it is difficult to come up with names for those operations.

(When coffee and sugar are put together in a cup we call it adding sugar to coffee. When a book is revised for the second edition the author may "add" a chapter to it. ) But conventionally the word "addition" is used in algebra for commutative binary operations.

Consider all functions of the form $ax+b$ with $a, b$ real numbers and $a\neq 0$. SUch collections do not form a group under addition ($3x-5$ and $-3x-5$ when "addded" gives the function $0x-5$ which is not in our set).

However this collection forms a group when we define "composition of functions" as the binary operation. Composition $(ax+b)\circ (cx+d) = cax + bc+d$. This is not equal to $(cx+d)\circ (ax+b)$, hence not abelian. (Check it). The inverse function for $ax+b$ would be $\frac 1a x -\frac ba$.