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IB Mathematics HL Option: Sets, Relations and groups (Abelian Group)

How can we show that the set under the addition is an Abelian group?

Thanks

How can we show that the set under the addition is an Abelian group?

Thanks

- elizabeth
**Posts:**0**Joined:**Mon Jan 28, 2013 8:09 pm

IB Mathematics HL Option Sets, Relations and Groups - Abelian Group

The 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

The 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

- miranda
**Posts:**268**Joined:**Mon Jan 28, 2013 8:03 pm

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