I was reading a post here that give some interesting approach about isomorphism theorem (see quote). But there are some things I don't understand. What exactly does this mean?
The Second Isomorphism Theorem says that the homomorphism F is the same on the restriction to H (by restricting the kernal) as it is on the smallest subgroup that contains both K and H. F cannot tell the difference between them.
And also the explanation for this
The Third Isomorphism Theorem tells us that making this stop into G' does not affect anything
is not so clear to me, what exactly does it mean by does not affect anything?
This was the original post :
The isomorphism theorems overall tell us that Homomorphisms and Normal Subgroups are essentially the same thing. Or, alternatively, Homomorphisms are classified by their Kernals.
The First Isomorphism Theorem sets up the correspondence: For every Normal Subgroup K, there is a "unique" surjective Homomorphism with kernal K. (I use quotes because we can compose with any isomorphism of G/K to get another, but we can see these as the same) Any other homomorphism with kernal K will contain the image of this homomorphism as an ordinary subgroup.
That is good and all, but we can do various things within G that may change what a homomorphism looks like. We want to classify some of these things.
If we have a homomorphism F from G with kernal K and a subgroup H of G, how does F behave when we restrict to H? First, we know that the kernal is going to be K∩H, simply by definition. But if we look at the subgroup KH, then we are still essentially restricting to K (because for kh in KH, F(kh)=F(h), so it only depends on K) but this new group contains H now. This can be see as the smallest subgroup of G that contains both K and H. The Second Isomorphism Theorem says that the homomorphism F is the same on the restriction to H (by restricting the kernal) as it is on the smallest subgroup that contains both K and H. F cannot tell the difference between them. You should view the Second Isomorphism Theorem as the Isomorphism Theorem of Function Restriction.
What happens if we compose two functions? If F is a surjective function from G to H, but we can factor this into a composition of two functions A from G to G' and B from G' to H (so F=B*A), then what happens to the kernals? The Third Isomorphism Theorem tells us that making this stop into G' does not affect anything. If N is the kernal of F, then the image of F is isomorphis to G/N. But if A has kernal K, then we need G'=G/K and the image of N is going to be N/K. Then going from G' to H is going to have kernal N/K and so H=G'/(N/K). You should see the Third Isomorphism Theorem as the Isomorphism Theorem of Function Composition.
It's a shame that the Isomorphism Theorems are all done in terms of quotient groups, which are dry and usually very unmotivated. The focus of an intro to Group Theory course should be on Homomorphisms and Group Actions rather than on group structure. It really helps motivate things, put them into context, apply more fluidly into other areas of math and you still get all the structure theorems you could ask for.
The First Isomorphism Theorem could be stated as such:
If K is a normal subgroup, then there is a homomorphism with kernal K and any two such homomorphisms have isomorphic images. Call the image G/K.
So we define G/K as the image of the homomorphism guaranteed by the First Isomorphism Theorem and then the proof of it would just be the typical construction of it. This really shows that the structure of G/K is determined as the image of a homomorphism instead of confusing things with cosets. Save cosets for Group Actions.