14.6. Definition of Cyclic Group

Try these first

1. 1.

   Define a cyclic group.

   Hint

   What element can generate all the others?

2. 2.

   Give an example of an infinite cyclic group.

   Hint

   Think about numbers and addition.

3. 3.

   What is a cyclic group?

   • A group generated by multiple elements
   • A group generated by a single element
   • A non-abelian group

   Hint

   Think about how many elements are needed to generate a group.

4. 4.

   True or False: Every finite group is cyclic.

   • True
   • False

   Hint

   Consider groups like S_3.

5. 5.

   Prove that the multiplicative group of integers modulo a prime is cyclic.

   Hint

   Consider the structure of numbers under modulo operations.

6. 6.

   Explain how to determine whether a given group is cyclic just by examining its elements and their orders.

   Hint

   Use the group's order and test with generators.