1. Chapter 4: Cyclic Groups
Properties of Cyclic Groups
Classification of Subgroups of Cyclic Groups

3. More Examples

4. More Examples
Therefore, U(8) is not cyclic.

6. Proof: continue

7. Proof: continue

9. If |a|=6

11. Proof; continue

12. Example:

19. A subgroup of a cyclic group is cyclic.
If |<a>|=n, then the order of any subgroup of <a> divides n.
For each +ve integer k, where
k divides n, the group <a> has exactly one subgroup of order k, namely,

20. Understanding the theorem

21. Proof: theorem 4.3

25. Example

29. Euler phi function

31. Example
Find the number of elements of order 8 in the cyclic group

33. Example: find the euler function value for each of the following numbers: n=81, n=100

34. The subgroup lattice