A finite field with four elements

 In the last post, we discovered that for any prime p, ℤp is a finite field, where the entire set under addition is an abelian group and ℤp-{0} is an abelian group under multiplication. On the other hand, if n is a composite number, ℤn-{0} is not a group under multiplication because there will be at least two numbers between 1 and n-1 that multiply to n, which in modulo n arithmetic means j*k = 0, and neither j or k will have multiplicative inverses.

Let us instead look at an additive group of four elements, 0, 1, a and 1+a. Here is the Cayley table for addition.

 

      0    1    a    1+a

0   | 0    1    a    1+a

1   | 1    0    1+a   a 

a   | a    1+a   0    1 

1+a | 1+a  a     1    0 

Every element is its own additive inverse, so this is isomorphic to the Klein four-group. What about the non-zero elements under multiplication? There are three such elements, so if this is a group, it must be isomorphic to ℤ3. Let's figure out the Cayley table.

 

      1    a    1+a

1   | 1    a    1+a

a   | a    ?    ?  

1+a | 1+a  ?    ?  

Recall that every element must show up in every row and column, so a*(1+a) must equal 1, and likewise (1+a)*a.

      1    a    1+a

1   | 1    a    1+a

a   | a    ?    1  

1+a | 1+a  1    ?  

The other two question marks can be sussed out by using the Latin square rule.

      1    a    1+a

1   | 1    a    1+a

a   | a    1+a  1  

1+a | 1+a  1    a  

In regular multiplication, (1+a)²= 1 + 2a + a², but because this group uses mod 2 arithmetic, this simplifies to 1 + a². On the table we see a² = 1+a, so we get 1+ 1+a, and mod 2 simplifies this to a. I leave it as an exercise for the reader to see that any combination of these four elements, will be consistent with our addition and multiplication tables.

 

In the literature, a field whose order is q = pⁿ is called Fq or 𝔽q or GF(q), which stands for Galois field.

 

Z_2 x Z_2, our first example of a direct product group

 If we have two groups, G and H, we can make a new group GxH, which is known as the direct product of G and H. The order of this group is the product of the orders of G and H. There are many ways to represent groups like this, one common method is to make all the possible ordered pairs of the for (g,h), where the coordinate is an element of G and the second an element of H.

Since the group (Z2,+) has the elements 0 and 1, we could represent Z2 x Z2 with four ordered pairs (0,0), (1,0) (0,1) and (1,1), and the operator would be addition modulo 2 for both coordinates.

I have used my natural mathematical laziness to write the group elements as 1, a, b and ab, and to make the group operator multiplication. When we have at least two symbols which are called the generators of the group, we need to explain their relationship.

The relationship of a and b is as follows.

a² = 1, b² = 1, ab = ba. I could have been even lazier and called ab by the letter c, but even my laziness has its limits.

There are three subgroups of order 2, and as always, just one subgroup of order 1, which is the identity element all by itself.

Another common name for this group is the Klein four-group, named for the 19th Century German mathematician Felix Klein.

 

 Klein's name is also attached to the Klein bottle, a mathematical curiosity that cannot truly exist in three dimensions. If you have ever studied the Möbius strip,you will know it has only one side and only one edge. The Klein bottle can be described as two
Möbius strips glued together edge to edge. In regular three dimensions, this is an impossible task, because the bottle must pass through itself without the benefit of having a gap to pass through. Just as a Möbius strip has only one, likewise a Klein bottle is one-sided, so there is no "inside" and "outside".

This may seem just like an odd distraction with no earthly purpose except to prove mathematicians are weird, but differential equation solution sets create manifolds, usually thought of as surfaces like the skin of a sphere or torus. The double pendulum, which has a famously chaotic movement pattern, does have a differential equation solution set and the shape of that manifold is... wait for it... 

 

a Klein bottle.

 

This video on YouTube shows some single pendulum actions, then at 0:49, shows the double pendulum motion.