14. Cyclic Groups

Try these first

1. 1.

   Define what a cyclic group is.

   Hint

   Think about how one element can create others.

2. 2.

   What is the identity element in a group?

   Hint

   Consider what element would keep others 'the same.'

3. 3.

   What is a cyclic group?

   • A group with no generators
   • A group that can be generated by a single element
   • A collection of elements without operations

   Hint

   Think about what defines the word 'cyclic'.

4. 4.

   True or False: In any group, the identity and inverse elements are unique.

   • True
   • False

   Hint

   Recall the properties of group definitions.

5. 5.

   Given a cyclic group of order 8 generated by g, list the elements of the group.

   Hint

   Start with the generator and calculate its powers.

6. 6.

   If a generator of a cyclic group has order 12, what can be said about the powers of this generator?

   Hint

   Consult your knowledge on the structure of cyclic groups.