13.6. Abstract Groups

Try these first

1. 1.

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

   Hint

   Think about what number added to any integer gives the same integer.

2. 2.

   True or False: The set of all even integers forms a group under addition.

   Hint

   Check if all group properties are satisfied.

3. 3.

   What are the properties that must be satisfied for a set to be a group?

   Hint

   List down each property.

4. 4.

   True or False: A set must always have a commutative operation to be considered a group.

   • True
   • False

   Hint

   Recall the definition of Abelian groups.

5. 5.

   Prove that the set of rational numbers excluding zero forms a group under multiplication.

   Hint

   Verify closure first.

6. 6.

   Create an abstract group operation that meets all the properties but is not traditional addition or multiplication.

   Hint

   Think outside numeric operations.