Chapter 2: Groups

Definition and Examples of Groups

Elementary Properties of Groups

Definition

Binary OperationLet G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G.

That is for each

abba

GabGba

),(

element an is there,

Definition : Group

Abelian Group A group G is called an Abelian Group ifab=ba for all elements a,b in G. G is called non Abelian Group if ab ≠

ba for some a,b in G.

Examples 1/

Examples 2/

Multiplication table for {1,-1,i,-i}-i i -1 1

-i i -1 1 1

i -i 1 -1 -1

1 -1 -i i i

-1 1 i -i -i

Examples 2/

Examples 3/

Examples

examples

examples

This is a non Abelian group

examples

examples The group U(n).

Note that U(p)={1,2,3,…,p-1} if p is prime

The following examples are not groups:

examples

The group SL(2,F)

Then SL(2,F) is a group under multiplication of matrices called

the special linear group.For example SL(2,Z5)

1.det(A)

and from entries with matrices 22 all ofset thebe

),2(let ,or Z ,, ofany be Let p

FA

FSLCRQF

The group GL(2, Z5)

In a group G, there is only one identityelement.