14.7. Examples of Cyclic Groups

Try these first

1. 1.

   Define a cyclic group.

   Hint

   Think about groups that can be fully defined by one number.

2. 2.

   What is the identity element in a group?

   Hint

   This element is critical for maintaining the structure of a group.

3. 3.

   What defines a cyclic group?

   • One generator can produce all elements.
   • Two identities exist.
   • No inverses are required.

   Hint

   Focus on the unique feature of cyclic groups.

4. 4.

   True or False: The identity element is unique in any group.

   • True
   • False

   Hint

   Consider the properties of identity in group theory.

5. 5.

   Prove that the group of integers under addition is cyclic.

   Hint

   Consider the operations performed by adding one repeatedly.

6. 6.

   Show that every element of Z/7Z can be generated by an appropriate base, and identify the generators.

   Hint

   Work modulo 7 and document each result.