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.