13.2.3. Identity Element

Try these first

1. 1.

   Identify the identity element for the group of integers under addition.

   Hint

   Think about what number you can add to any integer to get that integer back.

2. 2.

   Is the set of positive integers under addition a group? Why or why not?

   Hint

   Consider if you have '0' or a similar number in that set.

3. 3.

   What is the identity element of the group of integers under addition?

   • 1
   • 2
   • 0

   Hint

   Look at what happens when you add 0.

4. 4.

   True or False: Every group is Abelian.

   • True
   • False

   Hint

   Think about the order of operations in groups.

5. 5.

   Prove that the identity element is unique in any group.

   Hint

   Use the definitions of identity element properties in your proof.

6. 6.

   Consider a set of complex numbers under multiplication. Describe the identity element and prove its existence.

   Hint

   Apply the definition of multiplication for complex numbers.