Groups

A group is a set of elements closed to a binary operation and several required axioms are true.

We say that a set is closed to a binary operation if we can take any two elements from that set, perform the binary operation on those two elements, and the result is also an element of the set. The same element may be taken twice.

With all this in mind, let’s test the set {+1,-1} to see if it is closed to Multiplication.

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

All four of the possible multiplications result in an answer that is in the set. Thus, {+1, -1} is closed to Multiplication.

We can think of another finite set that forms a group with Multiplication:

{i, -i, 1, -1}

A group must have associativity, identity elements, inverse elements and closure for the operations.

For our first set, +1 is the identity element.

For inverse elements, each element is its inverse:

+1 * +1= +1

-1 * -1 = +1

In a way, this is bad because it doesn’t show an important idea, so let’s look for the inverse elements of {i, -i, 1, -1}.

The elements i and – i are inverse elements of each other.

-i*i=+1 (because i*i=-1)

Fot 1 and -1, its the same as before, the element is its own inverse since its square equals +1.

More work coming–how do we prove associativity of multiplication? Right now we say that addition is the aggregation of countings, and multiplication is the repetition of addition: 5*3 is +5+5+5 or “plus 5 three times”. Multiplying three values is building a 3D rectangle where each value is a length of one of the sides of the 3D rectangle and the answer to the multiplication is the volume of the 3D rectangle.

For (3*4)*5 = 3*(4*5)

• (3*4)*5 is five slices of (3*4)
• 3*(4*5) is three slices of (4*5)

We can generalize this to (a*b)*c – a*(b*c) and since we are slicing up the same 3D rectangle each time, the answer to the calculate of the volume of the the 3D rectangle will be the same.