## IB Maths HL Option Sets Cayley table

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 Cayley table

IB Mathematics HL Option: Sets, Relations and groups (Abelian Groups- Cayley table)

How can we show that the set $\mathbb{Z}_{3}$ under the addition modulo 3 is an Abelian group? and what is the order of each element?

Thanks
elizabeth

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Joined: Mon Jan 28, 2013 8:09 pm

### Re: IB Maths HL Option Sets Cayley table

IB Mathematics HL Option Sets, Relations and Groups - Cayley table

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., $x*y \in G$ for every $x,y \in G$

2. The operation * is associative on G,
i.e., $(x*y)*z=x*(y*z)$ for every $x,y \in G$

3. The operation * has an identity element in G,
i.e., $x*e=e*x=x$ for every $x \in G$

4. Each element of G has an inverse under *,
i.e., $x*x^{-1}=x^{-1}*x=e$ for every $x \in G$

A group {G,*} is called Abelian if $x*y=y*x$ for every $x,y \in G$

First we must construct the Cayley table in order to determine closure and the existence of identity and inverse elements.

$\begin{pmatrix} + & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 2 & 0 & 1 \end{pmatrix}$

the set $\mathbb{Z}_{3}$ under the addition modulo 3

Closure: The addition modulo 3 on the set $\mathbb{Z}_{3}$ is closed
i.e.$(x+y)\in \mathbb{Z}_{3}$ for every $x,y \in \mathbb{Z}_{3}$

Associative: The addition on $\mathbb{Z}_{3}$ is Associative
i.e. x+(y+z)=(x+y)+z for every $x,y \in \mathbb{Z}_{3}$

Identity: For every element $x \in \mathbb{Z}_{3}$ there is an element, the zero, which is the identity element in $\mathbb{Z}_{3}$ under addition.
i.e. x+0=0+x=0

Inverse: The identity element {0} appears once in every row and every column, so every element $x \in \mathbb{Z}_{3}$ has an iverse.

Commutative: The addition on $\mathbb{Z}_{3}$ is Commutative
i.e. x+y=y+x for every $x,y \in \mathbb{Z}_{3}$

As for the order you have to know the following
The order of a group {G,*} is the number of elements in G
The order of an element x of a group {G,*} is the smallest positive integer n for which $x^n=e$ with e the identity element.
A group {G,*} is said to be finite if it has a finite order and is said to be infinite if it has infinite elements.

So in you exercise the identity element has order 1 since 0=0
the element 1 has order 3 since 1+1+1=0
and finally the element 2 has also order 3 since 2+2+2=0

Hope these help
miranda

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