Vrftsjtryk

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.