14.8. Properties of Cyclic Groups

Try these first

1. 1.

   Define a cyclic group.

   Hint

   Think about the relationship between an element and the entire group.

2. 2.

   What is the identity element in a group?

   Hint

   Consider what you get when you operate with the identity.

3. 3.

   What is the defining characteristic of a cyclic group?

   • It can be generated by multiple elements
   • It can be generated by a single element
   • It has no identity element

   Hint

   Recall what a generator does.

4. 4.

   True or False: Every element in a finite cyclic group has a finite order.

   • True
   • False

   Hint

   Think about how the structure of finite groups differs from infinite ones.

5. 5.

   Given the group of integers modulo 7 with addition as the operation, discuss the generators and their power relations.

   Hint

   Consider how you can reach each element from your generators.

6. 6.

   If in a cyclic group of order 12, you find an element of order 4, what other orders are possible for elements in this group?

   Hint

   Think about how group orders relate to different elements.