Group Theory

A Group is a Set and a Binary Operation and there are four axioms that are true:

  1. Closure of the elements in the set to the binary operation
  2. An Identity Element exists
  3. For each element, its Inverse Element is also in the set
  4. Associativity is true for any possible combination of three elements: that is to say (a*b)*c=a*(b*c)

The simplest example we can think of is the set {-1, +1} and the operation *.

  • -1*-1=+1
  • -1*+1=-1
  • +1*-1=-1
  • +1*+1=+1

In the four math computations shown above, we can see that * on any two elements of the set results in an answer that is a member of the set, hence we have closure to *.

We won’t show all the possible combinations for (a*b)*c=a*(b*c) because we already know that associativity is true for the multiplication of numbers. In another example we’ll show the massive, gory details needed to prove Associativity.

Group Theory isn’t limited to numbers. Let’s change the game.

  • Define +4 as being such that on a clock it will take you from 12 to 4.
  • Define -4 as being such that on the clock it will take you from 12 to 8.

So far, in our set, we have {+4, -4}. Let’s use the symbol * for our operation.

Doing +4 and then another +4 takes us to 8. We recall that doing -4 once took us to 8.

(+4) * (+4) = (-4)

If you think about it, it should make sense that we can do something similar the other way:

(-4) * (-4) = (+4)

We’re almost done, now look at this:

(+4) * (-4) = ?

We need an operation that doesn’t take us anywhere, something that starts at 12 and finishes at 12. Or it starts at 4 and finishes at 4. The “don’t move” operation. Think back to adding 0 and multiplying by 1. It is tradition to use the symbol E for “don’t move anywhere” or “don’t change the value”. It is called the Identity Operation. If we add E to our set we can write the following:

  • (+4) * (-4) = E
  • (-4) * (+4) = E

We can now say that {-4, E, +4} is closed to *. The work to prove Associativity is shown in Appendix A.

We can add operations and axioms to a group, you get something more specific, such as a ring or a field. We speak of adding “structure” when we pile on more rules. Our motivation for piling on more structure is to make something generic that will work a necessary job elsewhere. For example, we build a generic structure for Fields and later we learn that a Vector Space needs to draw values from a Field.

Flip Game

To drive home further the ideas, lets take a penny and play a game. You don’t have to tell me whether you you have heads or tails, the operation r means to read whether the result is the same as a “nothing” or the same as a “flip”. Does the following make sense?

  • (flip)*(flip)=(same)
  • (flip)*(same)=(flip)
  • (same)*(flip)=(flip)
  • (same)*(same)=(same)

Thinking back to the first set that we examined, can you see the following?

  • A correspondence between “-1” and “flip”.
  • A correspondence between “+1” and “same”.

Possibly you might suddenly think of something else that fits this pattern, or months from now you come across something and it makes you think of this.

We say that {same, flip} and your new-found set share something in common with the Cayley Table below:

\begin{bmatrix} * && -1 && +1 \\ -1 && +1 && -1 \\ +1 && -1 && +1 \end{bmatrix}

Appendix A

We needed 27 calculations, each to confirm an equality, so a computer program was written to generate all that work, shown below:

(+4 * +4) * +4 = +4 * (+4 * +4)
-4 * +4 = +4 * -4
E = E

(+4 * +4) * -4 = +4 * (+4 * -4)
-4 * -4 = +4 * E
+4 = +4

(+4 * +4) * E = +4 * (+4 * E)
-4 * E = +4 * +4
-4 = -4

(+4 * -4) * +4 = +4 * (-4 * +4)
E * +4 = +4 * E
+4 = +4

(+4 * -4) * -4 = +4 * (-4 * -4)
E * -4 = +4 * +4
-4 = -4

(+4 * -4) * E = +4 * (-4 * E)
E * E = +4 * -4
E = E

(+4 * E) * +4 = +4 * (E * +4)
+4 * +4 = +4 * +4
-4 = -4

(+4 * E) * -4 = +4 * (E * -4)
+4 * -4 = +4 * -4
E = E

(+4 * E) * E = +4 * (E * E)
+4 * E = +4 * E
+4 = +4

(-4 * +4) * +4 = -4 * (+4 * +4)
E * +4 = -4 * -4
+4 = +4

(-4 * +4) * -4 = -4 * (+4 * -4)
E * -4 = -4 * E
-4 = -4

(-4 * +4) * E = -4 * (+4 * E)
E * E = -4 * +4
E = E

(-4 * -4) * +4 = -4 * (-4 * +4)
+4 * +4 = -4 * E
-4 = -4

(-4 * -4) * -4 = -4 * (-4 * -4)
+4 * -4 = -4 * +4
E = E

(-4 * -4) * E = -4 * (-4 * E)
+4 * E = -4 * -4
+4 = +4

(-4 * E) * +4 = -4 * (E * +4)
-4 * +4 = -4 * +4
E = E

(-4 * E) * -4 = -4 * (E * -4)
-4 * -4 = -4 * -4
+4 = +4

(-4 * E) * E = -4 * (E * E)
-4 * E = -4 * E
-4 = -4

(E * +4) * +4 = E * (+4 * +4)
+4 * +4 = E * -4
-4 = -4

(E * +4) * -4 = E * (+4 * -4)
+4 * -4 = E * E
E = E

(E * +4) * E = E * (+4 * E)
+4 * E = E * +4
+4 = +4

(E * -4) * +4 = E * (-4 * +4)
-4 * +4 = E * E
E = E

(E * -4) * -4 = E * (-4 * -4)
-4 * -4 = E * +4
+4 = +4

(E * -4) * E = E * (-4 * E)
-4 * E = E * -4
-4 = -4

(E * E) * +4 = E * (E * +4)
E * +4 = E * +4
+4 = +4

(E * E) * -4 = E * (E * -4)
E * -4 = E * -4
-4 = -4

(E * E) * E = E * (E * E)
E * E = E * E
E = E

The first check we did was to glance through the answers to see if each of the three possible answers appeared nine times.