13. Group Theory

Try these first

1. 1.

   What are the four axioms that define a group?

   Hint

   Use the acronym C-A-I-I.

2. 2.

   Give an example of a group under addition.

   Hint

   Think about the properties of integers.

3. 3.

   Which property ensures that the result of an operation on two elements is still in the set?

   • Closure
   • Associativity
   • Identity

   Hint

   Think about how operations interact with set membership.

4. 4.

   True or False: Every group operation must be commutative.

   • True
   • False

   Hint

   Recall the definition of an Abelian group.

5. 5.

   Prove that any finite subset of a group is also a group under the same binary operation, citing each of the group axioms.

   Hint

   Focus on the properties of the larger group.

6. 6.

   Given the set of integers modulo n, discuss how it forms a group with addition, relating it to all four axioms.

   Hint

   Revisit properties of modulo arithmetic.