14.4. Order of a Group Element

Try these first

1. 1.

   What is the identity element in a group?

   Hint

   Think about the role of 0 in addition or 1 in multiplication.

2. 2.

   Define the order of an element.

   Hint

   What do you get when you finally reach the group's starting point?

3. 3.

   What is the order of an element?

   • The smallest positive integer for multiplication
   • The highest exponent possible
   • A measure of the group size

   Hint

   Think about the cycle and repetition in operations.

4. 4.

   In a cyclic group, can you have multiple generators?

   • True
   • False

   Hint

   Consider the number of ways you can reach a full cycle.

5. 5.

   Prove that if a group has an element with infinite order, it cannot be finite.

   Hint

   Consider the implications of having a finite number of elements but extending indefinitely with one.

6. 6.

   Given a group (Z/7Z, +), describe its properties and the order of each element.

   Hint

   Think about how addition mod 7 would work, and apply the elementary properties.