Groups

A group is a combination of a nonempty Set and a Binary Operation to which there are attached four rules.

Failure to follow the four rules results in an umpire telling you that the set is not a group.

For our example we chose the nonempty set to be {-1, +1} and the operation to be multiplication (\cdot[latex]).</p> <p>Rule 1: The binary operation on any two elements must produce an element in the set. We can quickly test all possibilities:</p> <ol class="wp-block-list"><li>[latex]-1\cdot -1= +1

  • -1\cdot +1= -1
  • +1\cdot -1= -1
  • +1\cdot +1= +1

    • An Identity Element must exist. We examine the four equations above and notice that anytime +1 is involved the answer is the other thing, therefore +1 is the identity element.

    For each element 'a' in the group, there exist an Inverse Element '-a'. The binary operation on 'a' and '-a' results in the Identity element, which is often represented by 'e'.

    • a \cdot -a = e
    • -1 is the inverse element for -1
      • -1 \cdot -1 = +1
    • +1 is the inverse element for +1
      • +1 \cdot +1 = +1

    Associativity is true for any possible combination of three elements from the set.

    We already know from High School Algebra that we have Associativity for Addition and Multiplication.

    Appendix A

    Can you disqualify {-1, +1} and Addition {+} from being a group?

    Notice that +1 + +1 = +2 and +2 is not an element in the set. We don't have closure to addition.