13.3. Examples of Groups

Try these first

1. 1.

   What does closure mean in group theory?

   Hint

   Think about what happens when you add two integers.

2. 2.

   Identify the identity element in the group of integers with addition.

   Hint

   What number serves as a neutral element in addition?

3. 3.

   Which of the following is not a property of a group?

   • Closure
   • Distributivity
   • Identities

   Hint

   Focus on the fundamental group axioms.

4. 4.

   True or False: The set of natural numbers (excluding zero) is a group under addition.

   • True
   • False

   Hint

   Consider addition's identity and inverses.

5. 5.

   Demonstrate that the set of matrices of size 2x2 under addition forms a group.

   Hint

   Consider how matrices add together and whether their operations fit the group properties.

6. 6.

   Prove that the group of integers modulo n under addition isn't an abelian group if n is odd.

   Hint

   Consider how addition behaves under modulo conditions with odd-equal sums.