The invariants of conjugacy classes

 There is a direct link between groups represented by permutations and groups represented by permutation matrices. For example, if we have the permutation (12)(45) and we assume it belongs to S5, it can also be represented by the 5x5 matrix

0 1 0 0 0

1 0 0 0 0

0 0 1 0 0

0 0 0 0 1

0 0 0 1 0

In Sn, the conjugacy classes are all permutations with the same cycle structure, so the definition is necessary and sufficient. If we deal with subgroups of Sn, two conjugates must have the same cycle structure, but having the same cycle structure is not sufficient to state that the elements are conjugate. For example, the permutation (1234) generates a four element abelian group.

 

(1234)    

(1234)² = (13)(24)   

(1234)³ = (1432)

(1234)⁴ = (1)   

 

(1234) and (1432) have the same cycle structure, but this group is abelian and every element is in a conjugacy class by itself. In S4, (24)(1234)(24) = (1432), but (24) isn't available here.

 

A group of matrices also has invariants under conjugacy, the determinant, which was discussed in the last post and the trace, which is the sum of the elements along the main diagonal. Let's look at a four element group of 2x2 matrices that represent rotations of 0°, 90°, 180° and 270° in the xy-plane. This is an abelian group isomorphic to the earlier permutation group, so every conjugacy class is a singleton.

0° matrix

1  0 

0  1

determinant = 1, trace = 2

90° matrix

0 -1

1  0

determinant = 1, trace = 0

180° matrix

-1  0 

 0 -1

determinant = 1, trace = -2

270° matrix

0   1

-1  0

determinant = 1, trace = 0

Note that the 90° matrix and the 270° matrix have the same determinant and trace, which means there is a 2x2 non-singular matrix M such that M(90° matrix)M⁻¹ = 270° matrix. One such matrix is

 

-1  0

 0  1

 

which is not one of the elements of our defined group.

 

Commentary

 

As I have stated earlier, group theory is a generalization of the concept of symmetry. It is taught as an abstract field and the applications seem remote, with the possible exception of Rubik's Cube solutions. In fact, having symmetry in a physical problem makes it easier to solve. According to my professor Stu Smith, all solved differential equations rely on symmetry except for one. The differential equation that solves the solitary wave, also known as a soliton, does not have symmetry.

Most waves have peaks and valleys, and when the hit other waves, they can add to each other or cancel each other out, but a soliton only is almost all peak with a tiny valley. The bigger a soliton is, the faster it moves. In the ocean, we call the biggest solitons tsunamis.

 

A basic tenet of physics is big + fast = fuck you up. This is why a tsunami can wreak havoc when it hits land thousands of miles away from the source, because it will nearly the same speed and size it had when it was formed.

 

More on normal subgroups.

 We have a definition of a normal subgroup H contained in G, where for all x in G, xH = Hx. But why normal subgroups are important has not yet been discussed. Let me explain some things, proving some and stating others without proof.

1. if a is in H, all conjugates of a are also in H.

Proof. All conjugates of a can be written as xax⁻¹ for some x in G. Consider the set xHx⁻¹. We know xH = Hx, so we can rewrite our considered set as Hxx⁻¹, which simplifies to H. Since a is in H, every xax⁻¹ is also in H. 

 

Corollary. Every normal subgroup is a union of conjugacy classes.

 

The inverse truth of this says a non-normal subgroup H must have elements that do not have all their conjugates in H.

2. Every kernel of a homomorphism must be a normal subgroup.

Proof. Let f:G1→G2 be a homomorphism. If h is in the kernel of f, then f(h)f(x) = f(hx) = f(x), since f(h) is mapped to the identity of G2.  This does not mean hx = xh necessarily, but that Hx = xH, a subset of G1 that has as many elements as the order of H. A homomorphism splits the domain of f into a partition of equal sized sets called cosets.

 

Corollary: If G is a finite group and H is a subgroup, the order of H must divide the order of G.

For example, a set with 6 elements can only have subgroups of order 1 (the identity), 2, 3, or 6 (the whole group).

Definition: If H is a normal subgroup of G, the factor group G/H is a group created by an epimorphism f:G→G/H, where the co-domain is a group whose order is the order of G divided by the order of H.

An epimorphism is onto, so every element of G is mapped to a unique element of G/H. If G is abelian, then G/H must also be abelian, but if G is non-abelian, then G/H might be abelian or non-abelian.

 

In the next post, there will be examples of factor groups, some of which we have already seen.

 

Commentary

 

I am introducing topics in a completely different order than how I learned group theory from Ted Tracewell. It was a Monday/Wednesday/Friday class and on the first Friday, Tracewell started showing us examples of finite groups and subgroups. I went up to him after class, noticing a pattern that all the subgroups were of orders that divide the order of the group, and asking if it was a coincidence.

 

"No! It's always true! You have enough information to prove it yourself! Don't look it up!"

 

It was 1977, there was no easy access to all the world's knowledge, so I took the challenge. I finally got the proof to click on Sunday night after a lot of false starts. When I showed the proof to Tracewell on Monday, he asked what I called the equal sized subsets. I didn't have a special name for them, I just called them partitions.

 

If I had used the word "coset", he would have known I looked up the answer. My lack of knowledge of the term convinced him I had done the work myself. And that Monday long ago was when I turned into a math major.