Chapter 9: Normal Subgroups and Factor Groups

1. Chapter 9:Normal Subgroups and Factor Groups • Normal Subgroups • Factor Groups • Applications of Factor Groups • Internal Direct Groups

2. j Note that H is normal does not mean ah=ha . It means, ah=h’a and ha=ah’’ For some h’,h’’ in H.

3. Theorem

4. Examples;

5. Examples;

6. Examples;

8. Factor Groups

9. Proof:

10. Examples;

11. Examples; cont • Compute the order of elements in G/H

12. Examples; cont

13. Examples;cont Let G=A_4, H={1,2,3,4}. Then G/H={H,5H,9H}. The elements of G/H are: H={1,2,3,4}, 5H={5,6,7,8}, 9H={9,10,11,12}

15. Note that |3H|=? • |7H|=|9H|=?

16. Example

18. Example:

19. Some relations between G and G/H

21. Remarks

23. Example

24. Exercise 65

26. Proof; continues

28. Definition: Suppose H,K are subgroups of a group G. We define HK as: HK={hk : h in H and k in K} . Note that HK is not necesserely a subgroup of G.

29. Definition: We say G is the internal direct product of H and K and we write G=H x K if • H, K are normal subgroups of G. • G=HK • H K={e}

31. Note:

32. Examples