14.5. Properties of Order of a Group Element

Try these first

1. 1.

   Define the order of an element in a group. Give an example.

   Hint

   Think about how many times you need to apply the operation to get back to the identity.

2. 2.

   What is the identity element in a group?

   Hint

   Consider the element that allows you to act without changing value.

3. 3.

   What is the order of an element g in a group?

   • 1. Largest integer such that g^n = g
   • 2. Smallest positive integer n such that g^n = e
   • 3. The total number of elements in the group

   Hint

   Think about what it means for an element to return to the starting point.

4. 4.

   True or False: An element with infinite order will eventually repeat itself.

   • True
   • False

   Hint

   Consider the definition of infinite versus finite orders.

5. 5.

   Prove that if in a finite cyclic group the orders of two distinct elements are equal, then those two elements must generate the same subgroup.

   Hint

   Think about how many distinct elements can be produced by the powers of each element.

6. 6.

   Demonstrate the necessity of the condition that the order of a generator must match the order of the cyclic group it generates.

   Hint

   Reflect on what it means to cycle back through generated elements.