1. Cyclic Groups

2. Definition G is a cyclic group if G = for some a in G.

3. Cyclic groups may be finite: In Z 4, = {1, 2, 3, 0} = Z 4 Z 4 is cyclic. In Z 5, = {1,2,3,4,0} = Z 5 Z 5 is cyclic In Z n, = {1,2,3,…,(n-1),0} = Z n Z n is cyclic for n ≥ 1.

4. Cyclic groups may be infinite. In Z, = In Q * = In GL(2,R) { … -2, -1, 0, 1, 2, …} { … -4, -2, 0, 2, 4, …} { … 1/4, 1/2, 1, 2, 4, …}

5. Quick Facts: Cyclic Groups Every cyclic group is Abelian. Not every Abelian group is cyclic. If H ≤, then H = where m is the smallest positive integer with a m in H. Every subgroup of a cyclic group is cyclic. Let G = with |G|=n The order of each subgroup of G divides n. There is exactly one subgroup of G with order k, namely,. Cyclic groups are the building blocks of all Abelian groups.

6. Group G *mod 65

7. Group G *mod 65 Reordered as

8. Group G Listing only powers