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

More Examples

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

Proof: continue

Proof: continue

If |a|=6

Proof; continue

Example:

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,

Understanding the theorem

Proof: theorem 4.3

Example

Euler phi function

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

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

The subgroup lattice