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.

 

Personal commentary

 In the Winter Quarter of 1977, just a few days after my 21st birthday, I started attending a class in Abstract Algebra, taught by Ted Tracewell at Cal State Hayward, now known as Cal State East Bay. Sad to say, I can find no pictures of him, but the celebrities who look most like him are the science fiction author Isaac Asimov without the muttonchops, or the songwriter/playwright/actor Adolph Green with glasses.

 

 

 

I just turned 66, and I now can identify that class as the one that changed my life. I show pictures of Asimov and Green smiling because that is how I remember Ted Tracewell. He had a real excitement about higher mathematics and it always shown through. When I taught, I tried to emulate that enthusiasm when I could, and some students commented on it. Some said they hated math, but because they saw I loved it, it made it easier to get through the class.

 

Many topics in group theory feel like solving puzzles, both the proofs and the exercises, like filling in Cayley tables and finding conjugacy classes, which I have shown on the blog. More puzzle-like topics include representing finite abelian groups, representing groups pictorially in terms of their generators the structure of normal groups and filling in character tables, the last topic part of group representation theory. The inter-connectivity of group theory makes it hard for me to figure out the order in which topics should be introduced.

I will often take side trips to talk about the lives of the mathematicians who made important discoveries, but today it's my personal history, important to me, if no one else.

Conjugates and conjugacy classes

 I foolishly thought I would be able to write one post every day on this blog, but that ain't happening. I have long debates with myself about the order of the topics, and the order in which I am presenting the topics is nothing like the order of Ted Tracewell's class in the 1970s. No disrespect to Tracewell, I loved him dearly as a teacher and a person. He invited me to dinners at his house, his wife was a great cook, and he introduced me at least once to strangers as "his son". Many math professors at Cal State Hayward called me , and in all modesty, I was a beloved student.

So many topics fold back on themselves in group theory. Its structure is like a complicated watch, and the topics are interconnected in remarkable ways. After learning the basic structure of groups and learning about non-abelian groups where A*B ≠ B*A in all cases, Tracewell taught us about group homomorphisms. Instead, I am going to talk about conjugacy and conjugacy classes first, and I will get to homomorphisms tomorrow, and will likely continue on homomorphisms for the rest of the week.

If you learned the quadratic formula, you should have been taught about complex numbers of the form a∓bi, where i is the imaginary number that solves the equation x² = -1. If the discriminant b²-4ac is negative, the square root in the numerator is imaginary, a multiple of i. The complex numbers a+bi and a-bi are complex conjugates. They are not related to conjugates in group theory, so I wanted to make that clear. Along the same lines, degrees of freedom and degrees in geometry are not related, so I always tried to make that clear when degrees of freedom were introduced in class.

If a = xbx⁻¹, then a and b are conjugates in a group.

 

If a and b are conjugates in a group, they belong to the same conjugacy class.

Let me go through some of the basics.

1. The identity e is always in a conjugacy class of its own. The reason for this is e commutes with everything so

xex⁻¹ = exx⁻¹ = e.

 

2. If an element a commutes with all elements in a group G, then a is in a conjugacy class of its own. The reasoning is the same as above.

xax⁻¹ = axx⁻¹ = a.

 

3. In any abelian group, every conjugacy class contains exactly one element, and such classes are known as singletons.

 

4. Only in non-abelian groups do we have conjugacy classes with more than one element.

We have only learned so far about one non-abelian finite group, D4, so that will be our example at the end of the post.

5. Conjugacy is a relation that is reflexive, symmetric and transition, and having those three properties means the relation produces equivalency classes, splitting the group G into disjoint sets where the union of all those sets is G itself.

Reflexive. All elements are conjugate to themselves, because eae = a.

 

Symmetric: if b = xax⁻¹, then x⁻¹bx = x⁻¹xax⁻¹x = a.

 

Transitive. if b = xax⁻¹ and c = yby⁻¹, then c = yxax⁻¹y⁻¹, and the inverse of yx is x⁻¹y⁻¹.

 

Now let's look at the conjugacy classes of the symmetries of the square D4. Here, because this table is based on matrices, the identity is called I instead of e.

 

 

 

 

 

 

 

The I stands alone, so our first conjugacy class is {I}.

 

If you inspect the row for R180° and the column for R180°, you will see that R180° commutes with everything, so the second conjugacy class is {R180°}.

 

The two remaining rotations do not commute with the mirrors. We need to do all the possible version of Mx°R90°Mx° to prove my next statement, but I will skip to the result, our third conjugacy class is {R90°,R270°}.

 

And now we need to find the conjugacy classes for the Mx° elements. Again, the reader should check my work, but you will find there are two more conjugacy classes, {M90°,M0°} and {M45°,M135°}.

 

We have seen how to represent non-abelian groups as matrices and as permutations. We will see that conjugacy classes have nice properties in both representations.

 

Tomorrow: Our first look at group homomorphisms, mapping one group onto another in a way that preserves group operations.