13.6.2. Group Order

Try these first

1. 1.

   Define the closure property in your own words.

   Hint

   Think about adding two integers.

2. 2.

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

   Hint

   Consider what number you can add to any integer without changing its value.

3. 3.

   Which of the following is NOT a group axiom?

   • Closure
   • Associativity
   • Commutativity
   • Inverse

   Hint

   Remember that not all operations have to be commutative.

4. 4.

   True or False: The identity element of a group must be unique.

   • True
   • False

   Hint

   Consider how identity relates to every other element.

5. 5.

   Given the set of rational numbers excluding zero under multiplication, prove or disprove that it forms a group.

   Hint

   Establish closure by multiplying two rationals and see if the result is rational.

6. 6.

   Construct a finite group using a set of elements {0, 1, 2} under addition modulo 3, and verify all group properties.

   Hint

   Perform addition and standard mod operations for verification.