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

2. 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

3. Definition : Group

4. 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.

5. Examples 1/

6. Examples 2/

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

8. Examples 2/

9. Examples 3/

10. Examples

11. examples

12. examples
This is a non Abelian group

13. examples

14. examples
The group U(n).
Note that
U(p)={1,2,3,…,p-1} if p is prime

15. The following examples are not groups:

16. examples

17. 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)

18. The group GL(2, Z5)

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