Chapter 2: Groups Definition and Examples of Groups Elementary Properties of Groups

Definition Binary Operation Let 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

Definition : Group

Abelian Group A group G is called an Abelian Group if ab=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}

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)

The group GL(2, Z5)

In a group G, there is only one identity element.