Even more about normal subgroups and examples of factor groups

Normal subgroups are an important part of the structure of a group. We already have the definition.

 

If H is a subgroup contained in G and for every element x in G, xH = Hx, then H is normal.

 

We have also defined G/H as a new group called the factor group.

 

Every group G that has more than one element must have at least two subgroups, G itself and the identity element e by itself. Both of these subgroups are normal. G/G is a one element group, which means (e,*) and G/e = G.

Lemma: If G has order 2n and H has order n, H must be normal and G/H is isomorphic to Z2.

Proof: The elements of G that are not in H are denoted as G-H. If h is in H, hH = Hh = H, because the subgroup is closed under the group operation. If x is in G-H, xH = Hx = G-H.

Every group of order 2 is isomorphic to Z2 because if we make the Cayley table, it is a 2x2 matrix with two elements that follows the rule of Latin square, that every row and every column has exactly one copy of each element.

+ | 0 | 1 

0 | 0 | 1

1 | 1 | 0

We have seen that (nZ, +) is a subgroup of Z. Z/nZ = Zn. This factor group has n infinite cosets we can call nZ, nZ+1, nZ+2,... up to nZ+(n-1). We take an element from each coset and make it the representative. Usually the representatives are the integers from 0 to n-1, but on a clock, for example, we use 12 instead of 0 in Z12.

 

What does (R/Z, +) look like? You can think of it as the half closed, half open interval [0, 1), or you could visualize the fact that 1 is mapped onto 0, so it is a closed loop. 

 

What does (Q/Z, +) look like? It looks like (R/Z, +) to the naked eye, and no amount of magnification will change that, but there are an infinity of numbers missing from the interval.

 

How about (R/Q, +)? This one is trippy. All we would see is the single point 0, but the irrationals would all be sent to an infinite number of infinitesimal values. If you want to learn more about the topic, follow this link to the Wikipedia page.   

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.

Normal Subgroups

 A subgroup H of a group G is normal if for every g in G, gH=Hg.

 

Normal subgroups are an important topic and I will try to go through this slowly. Some hopefully obvious observations first.

1. The whole group G is a normal subgroup of G.

2. The identity element by itself is a normal subgroup of G.

3. If G is abelian, then every subgroup of G is normal.

We will only find non-normal subgroups in non-abelian groups, and the only one I have presented so far is D4. Let's look at the Cayley table.

 

There are several normal subgroups in D4, including the subgroup of all the rotations {I, R90°, R180°, R270°}. In everyday language, if you rotate a rotation, you will get another rotation, and if you rotate a mirror, you will get another mirror.

There are four subgroups of order 2 that are not normal, and each one consists of the identity I and one mirror. Let's look at the group {I, M90°}. Some elements will give us gH = Hg, but all we need is one element to screw up and that proves the group is not normal. On the list below, the elements that don't work are written in red. If you want to check my work, multiplying on the left is taking entries from the column of that element and multiplying in on the right is taking the entries from the row.

I{I, M90°} = {I, M90°}, and {I, M90°}I = {I, M90°}

R90°{I, M90°} = {R90°, M135°}, and {I, M90°}R90° = {R90°, M45°}

R180°{I, M90°} = {R180°, M90°}, and {I, M90°}R180° = {R180°, M0°}

R270°{I, M90°} = {R270°, M45°}, and {I, M90°}R270° = {R90°, M135°}

M0°{I, M90°} = {M0°, R180°}, and {I, M90°}M0° = {M0°, R180°}

M45°{I, M90°} = {M45°, R270°}, and {I, M90°}M45° = {M45°, R270°}

M90°{I, M90°} = {M90°, I}, and {I, M90°}M90° = {M90°, I}

M135°{I, M90°} = {M135°, R90°}, and {I, M90°}M135° = {M135°, R90°}

 

In D4, there are two elements that commute with everything, I and R180°. All the rotations commute with each other, and likewise mirrors commute with mirrors. 

 

In any group G, the set of elements that commute with all other elements is called the center and is denoted as Z(G). The Z comes from the German word for center, zentrum. The center is always a normal subgroup. Proving it is a subgroup is left as an exercise for the reader and the answer will be in the comments tomorrow.