# An exercise with the third isomorphism theorem in group theory

• Alex Langevub
In summary: As you can see, having some lines makes it easier to check the steps, and I forgot to write ##k''##, I got it right in the end. You don't need that much in an exam, but you shoudl be clear on what you are doing and why.
Alex Langevub

## Homework Statement

Let ##G## be a group. Let ##H \triangleleft G## and ##K \leq G## such that ##H\subseteq K##.

a) Show that ##K\triangleleft G## iff ##K/H \triangleleft G/H##

b) Suppose that ##K/H \triangleleft G/H##. Show that ##(G/H)/(K/H) \simeq G/K##

## Homework Equations

The three isomorphism theorems.

## The Attempt at a Solution

a) Let ##k \in K##, ##h \in H## and ##g \in G##

##(gH)(kH)(gH)^{-1}## need to belong to ##kH##
I am not quite sure how to go about proving this.

b) Since ##K/H \triangleleft G/H##, we know that ##K\triangleleft G## as proven in a).

We can therefore apply the third isomorphism theorem which states that since ##H## and ##K## are both normal in ##G## and that ##H## is a subset of ##K##,
$$(G/H)/(K/H) \simeq (G/K)$$

Alex Langevub said:

## Homework Statement

Let ##G## be a group. Let ##H \triangleleft G## and ##K \leq G## such that ##H\subseteq K##.

a) Show that ##K\triangleleft G## iff ##K/H \triangleleft G/H##

b) Suppose that ##K/H \triangleleft G/H##. Show that ##(G/H)/(K/H) \simeq G/K##

## Homework Equations

The three isomorphism theorems.

## The Attempt at a Solution

a) Let ##k \in K##, ##h \in H## and ##g \in G##

##(gH)(kH)(gH)^{-1}## need to belong to ##kH##
I am not quite sure how to go about proving this.
You will have to use that ##K## is normal in ##G##, so ##h^{-1}kh=k'## and you can always write ##H=Hh^{-1}##. Proceed elementwise, just that you cannot assume the same ##h## as representative of ##H##. Every occurance has another ##h_i\,.## And don't forget the other direction!
b) Since ##K/H \triangleleft G/H##, we know that ##K\triangleleft G## as proven in a).

We can therefore apply the third isomorphism theorem which states that since ##H## and ##K## are both normal in ##G## and that ##H## is a subset of ##K##,
$$(G/H)/(K/H) \simeq (G/K)$$
I thought part b) is exactly the isomorphism theorem, which has to be shown, but if you have it already, then yes, your reasoning is correct.

Alex Langevub
fresh_42 said:
You will have to use that ##K## is normal in ##G##, so ##h^{-1}kh=k'## and you can always write ##H=Hh^{-1}##. Proceed elementwise, just that you cannot assume the same ##h## as representative of ##H##. Every occurance has another ##h_i\,.## And don't forget the other direction!

I thought part b) is exactly the isomorphism theorem, which has to be shown, but if you have it already, then yes, your reasoning is correct.

I am still unsure how to go about resolving this problem. I haven't really seen any examples of problems with quotients like this one. So it's the possible manipulations that I am unsure about. Please tell me if this is alright.This problem is an iif so I need to demonstrate both directions ⇐) and ⇒).⇒)
If ##K\triangleleft G##, so we have that ##g_1^{-1}k_1g_1 \in K ##.

$$(g_1h_1)(k_2h_2)(g_1h_1)^{-1}$$
lets set ##k_2 = g_1^{-1}k_1g_1##
$$(g_1h_1g_1^{-1})k_1(g_1h_2h_1^{-1}g_1^{-1})$$

I am not sure where to go from there. Or even if I am on the right track...

Let me see.

##(gh_1)(kh_2)(g^{-1}h_3)=gh_1kh_1^{-1}h_1h_2g^{-1}h_3=gk\,'h_1h_2g^{-1}h_3=gk\,'g^{-1}gh_1h_2g^{-1}h_3=k\,''gh_1h_2g^{-1}h_3 =kh_4h_3 \in KH## because both ##H## and ##K## are normal.

Now the other direction: ##K/H \trianglelefteq G/H \Longrightarrow K \trianglelefteq G\,.##

I assume you meant ##kh_4h_3 \in K/H## and not ##kh_4h_3 \in KH## at the end there.Now,
##\Leftarrow##

In an exam I would put a lot more effort into initializeg the different variables ##k_i##, ##g_i## and ##h_i## and justifying each step.

If ##K/H \triangleleft G/H##, we have that
##(gh_1)(k_1h_2)(g^{-1}h_3) = k'h'##
##gh_1k_1g^{-1}(gh_2g^{-1})h_3 = k'h'##
##g(h_1k_1)g^{-1}(h_4)h_3 = k'h'##
##g(k_2)g^{-1}h_5 = k'h'##
##\Rightarrow gk_2g^{-1} \in K##
##\Rightarrow K\triangleleft G##

Alex Langevub said:
I assume you meant ##kh_4h_3 \in K/H## and not ##kh_4h_3 \in KH## at the end there.
No. I meant what I wrote, except that I forgot to write ##k''## instead of ##k##. We still have elements: ##(gh_1)(kh_2)(g^{-1}h_3)=k''h_4h_3 \in KH## but from that passing to cosets yields ##[g][k][g]^{-1}=[k''] \in K/H## what we needed.
Now,
##\Leftarrow##

In an exam I would put a lot more effort into initializeg the different variables ##k_i##, ##g_i## and ##h_i## and justifying each step.

If ##K/H \triangleleft G/H##, we have that
Let me fit in some lines which helps me to sort stuff out. We have ##H \triangleleft G## and thus also ##H \triangleleft K##.
Now we have ##(gH)(k_1H)(g^{-1}H) \in K/H\,,## so
##(gh_1)(k_1h_2)(g^{-1}h_3) = k'h'##
##gh_1k_1g^{-1}(gh_2g^{-1})h_3 = k'h'##
##g(h_1k_1)g^{-1}(h_4)h_3 = k'h'##
##g(k_2)g^{-1}h_5 = k'h'##
##\Rightarrow gk_2g^{-1} \in K##
##\Rightarrow gk_2g^{-1} = k'h_6 \in KH \subseteq K##
##\Rightarrow K\triangleleft G##
O.k.

## 1. What is the third isomorphism theorem in group theory?

The third isomorphism theorem in group theory states that if G is a group, H and K are normal subgroups of G with H contained in K, then the quotient groups G/H and K/H are isomorphic.

## 2. How is the third isomorphism theorem different from the first and second isomorphism theorems?

The first and second isomorphism theorems deal with the structure of a group G in relation to its subgroups, while the third isomorphism theorem focuses specifically on the relationship between normal subgroups of G.

## 3. What is the significance of the third isomorphism theorem in group theory?

The third isomorphism theorem allows us to better understand the structure of groups and how they relate to each other. It also has many practical applications in various fields of mathematics and science, such as cryptography and quantum mechanics.

## 4. How is the third isomorphism theorem used in real-world problems?

The third isomorphism theorem is used to solve problems involving group structures, such as finding the order of a quotient group or determining if two groups are isomorphic. It is also used in various areas of mathematics, including algebra, number theory, and geometry.

## 5. Are there any limitations or exceptions to the third isomorphism theorem?

There are certain conditions that must be met for the third isomorphism theorem to be applicable, such as having normal subgroups and a group structure that follows certain rules. Additionally, there may be specific cases where the theorem does not apply, but these are usually uncommon and can be addressed using other techniques in group theory.

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