14.3. Group Exponentiation

Try these first

1. 1.

   Define the identity element.

   Hint

   Think about what happens when something added to 0 or multiplied by 1.

2. 2.

   What does it mean for an element to be a generator?

   Hint

   What structure would you use to generate other elements?

3. 3.

   What is the identity element in group theory?

   • An element that changes others
   • An element that does not change others
   • An element that has no inverse

   Hint

   Think about what happens when combined with other elements.

4. 4.

   True or False: Every group can have more than one identity element.

   • True
   • False

   Hint

   Consider the definition of identity.

5. 5.

   In a cyclic group of order 12, what is the order of the generator? Prove your reasoning.

   Hint

   Consider how many unique group elements can be formed by repeated application of the generator.

6. 6.

   For group G with elements g, g², g³ where g is a generator, determine if g⁴ can be expressed in terms of the existing elements.

   Hint

   Check if g raised to a high exponent remains in the scope of the original elements.