IB Maths HL Option Sets Abelian Group

Operation table of a group, Symmetries of plane figures, Cycle notation for permutations, Left and right cosets, group homomorphism, kernel, kernel and range of a homomorphism, isomorphisms

IB Maths HL Option Sets Abelian Group

Postby elizabeth » Thu Mar 07, 2013 7:45 pm

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
elizabeth
 
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Re: IB Maths HL Option Sets Abelian Group

Postby miranda » Thu Mar 07, 2013 7:59 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 :) :)
miranda
 
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