13.3.4. Non-zero Integers under Multiplication

Try these first

1. 1.

   Define a group in abstract algebra.

   Hint

   Think of things that need to be satisfied for a set to be considered a group.

2. 2.

   What is the identity element for multiplication?

   Hint

   It's the number that doesn't change other numbers when multiplied.

3. 3.

   What property fails for non-zero integers under multiplication?

   • Closure
   • Associativity
   • Identity
   • Inverse

   Hint

   Think about what happens when you try to find the inverses of integers.

4. 4.

   The multiplication of two non-zero integers results in a:

   • True
   • False

   Hint

   Recall the definition of closure.

5. 5.

   Demonstrate whether the set of rational numbers under multiplication forms a group. Describe which properties hold and which do not.

   Hint

   Consider testing each of the four properties explicitly.

6. 6.

   Explore how the operation of addition affects the structure of the set of integers and explain why it qualifies them as a group.

   Hint

   Provide examples for each property.