IB Mathematics HL Option Sets, Relations and Groups - Abelian GroupThe basic principles about Abelian Groups are:

First of all I remind you the definition of

Group (G,*)Let G be a non-empty set on which a binary operation * is defined . We say G is a Group under this operation if each of the following four properties hold:

1. G is

closed under * , i.e.,

for every

2. The operation * is

associative on G,

i.e.,

for every

3. The operation * has an

identity element in G,

i.e.,

for every

4. Each element of G has an

inverse under *,

i.e.,

for every

A group {G,*} is called

Abelian if

for every

Also Given a group {G,*} we have the following properties:

Left cancellation law: If x*y=x*z then y=z

Right cancellation law: If y*x=z*x then y=z

Now concerning your question,

The set

under the addition is an

Abelian group since

Closure: The addition on

is closed

i.e.

for every

Associative: The addition on

is Associative

i.e. x+(y+z)=(x+y)+z for every

Identity: For every element

there is an element, the zero, which is the identity element in

under addition.

i.e. x+0=0+x=0

Inverse: For every element

there is an element, which is its opposite in

.

i.e. x+(-x)=(-x)+x=0

Commutative: The addition on

is Commutative

i.e. x+y=y+x for every

Hope these help