How can we find all the normal subgroups?

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Sorry for answering your question with a question, but I need some help to understand your question.)In summary, we want to find all the normal subgroups of $D_n$, which is defined as $\langle a, s \mid a^2=1=s^n, asa=s^{-1}\rangle$. In order to do so, we need to pick 1 or 2 elements and see which group they generate, and then determine if it is a normal subgroup. We have three main types of subgroups: those generated by two rotations, those generated by a reflection and a rotation, and those generated by two distinct reflections. We also need to consider the cases where $n$ is even or odd, as well as the prime
  • #1
mathmari
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Hey! :eek:

I want to find all the normal subgroups of $D_n$.

We have that $$D_n=\langle a, s \mid a^2=1=s^n, asa=s^{-1}\rangle$$

We have that $K$ is a normal subgroup of $D_n$ iff $$gkg^{-1}\in K, \forall g\in D_n \text{ and } \forall k\in K$$

Could you give me some hints how we could find all these subgroups? (Wondering)
 
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  • #2
mathmari said:
Hey! :eek:

I want to find all the normal subgroups of $D_n$.

We have that $$D_n=\langle a, s \mid a^2=1=s^n, asa=s^{-1}\rangle$$

We have that $K$ is a normal subgroup of $D_n$ iff $$gkg^{-1}\in K, \forall g\in D_n \text{ and } \forall k\in K$$

Could you give me some hints how we could find all these subgroups? (Wondering)

Hey mathmari! (Smile)

How about enumerating all subgroups and see where that brings us? (Wondering)
To do so, we need to pick 1 or 2 (or more) elements and see which group it generates, and then see what it takes to make it normal.
 
  • #3
I like Serena said:
How about enumerating all subgroups and see where that brings us? (Wondering)
To do so, we need to pick 1 or 2 (or more) elements and see which group it generates, and then see what it takes to make it normal.

Do we maybe pick the elements $as^i$ and $s^j$ for $0\leq i,j <n$ ? (Wondering)
 
  • #4
Try finding all the cyclic subgroups, first.

With dihedral groups it is helpful to remember there are two types of elements: reflections, and rotations. The reflections have an "$a$" in them.

For subgroups generated by two elements, we have three main kinds of subgroups:

1. Generated by two rotations. This type should be trivial to analyze.
2. Generated by a reflection, and a rotation. This should be easy once you see what happens:

Hint: $s^ka = as^{-k}$, for all $k \in \Bbb N$. Use induction to prove this.

3. Generated by two distinct reflections. This one is a bit tricky.

General warning: it matters whether or not $n$ is even or odd (the even $n$'s produce more subgroups). The prime factorization of $n$ also matters.
 
  • #5
Deveno said:
Try finding all the cyclic subgroups, first.

Is $\langle s^i\rangle$ for $i\mid n$ a subgroup of $D_n$ ? (Wondering)

If it is, do we have to check if an element of the form $s^ka$ belongs to this subgroup or if it belongs to or generate an other subgroup? (Wondering)
Deveno said:
For subgroups generated by two elements, we have three main kinds of subgroups:

1. Generated by two rotations. This type should be trivial to analyze.
2. Generated by a reflection, and a rotation. This should be easy once you see what happens:
3. Generated by two distinct reflections. This one is a bit tricky.

1. Is it generated by a "$s$" ?
2. Is this subgroup of the form $\langle a, s\rangle$ ?
3. Is it generated by an "$a$" ?

(Wondering)
 
  • #6
I think you're working too hard. As you showed previously, any subgroup of $<s>$ is normal since $<s>$ is normal. So the normal subgroups are the identity, $D_n$, any subgroup of $<s>$ and maybe some more. So suppose $H$ is a proper normal subgroup that is not a subgroup of $<s>$. Then $H$ contains an element of the form $s^ka$ for some $k$. Compute the conjugate $$s^{-1}s^kas=s^ks^{-1}as=s^ka(as^{-1}a)s=s^kas^2\in H$$
So $s^2\in H$. If $n$ is odd, $K=<s^2>=<s>\subset H$ and so $H=D_n$. Suppose $n$ is even. Consider the factor group $D_n/K$. Easily then $H/K$ is one of the 3 proper normal subgroups of $D_n/K$. Thus $H=<s^2,a>$ or $H=<s^2,sa>$, each of these is isomorphic to $D_m$ where $m=n/2$.
 
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  • #7
johng said:
So suppose $H$ is a proper normal subgroup that is not a subgroup of $<s>$. Then $H$ contains an element of the form $s^ka$ for some $k$. Compute the conjugate $$s^{-1}s^kas=s^ks^{-1}as=s^ka(as^{-1}a)s=s^kas^2\in H$$
So $s^2\in H$.

Why does this stand? Why does $H$ contain an element of the form $s^ka$ ? (Wondering),

How do we conclude that $s^kas^2\in H$ ? (Wondering)
 
  • #8
$D_n=\{s^k\,:\,0\leq k<n\}\cup\{s^ka\,:\,0\leq k<n\}$. So if $H$ is not a subgroup (subset) of $<s>$, H contains an element of the form $s^ka$.

By definition of normality of $H$, any conjugate of an element of $H$ is also an element of $H$. So $s^kas^2\in H$ implies $s^2=(s^ka)^{-1}s^kas^2\in H$
 
  • #9
Let $H$ be a normal subgroup of $D_n$.

  1. If all the elements of $H$ are of the form $s^k$ then $H=\langle s^d\rangle$ for some $d$ that divides $n$.

    $\langle s^d\rangle$ is indeed a normal subgroup of $D_n$:
    It suffices to show that $gs^dg^{-1}\in \langle s^d\rangle, \forall g\in D_n$.
    • if $g=s^k$, then $s^ks^ds^{-k}=s^{k+d-k}=s^d\in \langle s^d\rangle$
    • if $g=as^k$, then $as^ks^d(as^k)^{-1}=as^ks^ds^{-k}a^{-1}=as^ks^ds^{-k}a=as^{k+d-k}a=as^da=a\underset{d \text{ times } "s"}{\underbrace{saasaas\dots as}a}=(asa)^d=(s^{-1})^d=s^{-d}\in \langle s^d\rangle$
  2. If $H$ contains an element of the form $s^ka$, for some $k$, then since it is a normal subgroup of $D_n$, it must hold that $$s^{-1}s^kas\in H \Rightarrow s^{-1+k}as\in \Rightarrow s^ks^{-1}as\in H \Rightarrow s^kas^2\in H$$

    So $s^2=(s^ka)^{-1}s^kas^2\in H$.

    But how can we find exactly which this $H$ is? (Wondering)
 
  • #10
I'm not exactly sure what your question is. Maybe this is what you're asking:
Assume $s^ka\in H$. If $k$ is even, say $k=2j$, then since $<s^2>\subseteq H$, $a=s^{-2j}s^{2j}a\in H$ and so $H=<s^2,a>$. Similarly, if $k$ is odd, $H=<s^2,sa>$.
 
  • #11
johng said:
Assume $s^ka\in H$. If $k$ is even, say $k=2j$, then since $<s^2>\subseteq H$, $a=s^{-2j}s^{2j}a\in H$ and so $H=<s^2,a>$.

Why does it stand that $<s^2>\subseteq H$ ? (Wondering)

Having shown that $s^ka\in H$ and $a\in H$, why does it imply that $H=<s^2,a>$ ? (Wondering)

How do we know that these two elements generate a subgroup? (Wondering)

johng said:
Similarly, if $k$ is odd, $H=<s^2,sa>$.

Assume $s^ka\in H$. When $k$ is odd, say $k=2j+1$, does it stand that $<s^2>\subseteq H$ ? (Wondering)
If it holds, then $sa=ss^{-2j-1}s^{2j+1}a=s^{-2j}s^{2j+1}a\in H$ and so $H=<s^2,sa>$, right? (Wondering)
 
  • #12
$$D_n=\langle a,s\mid s^n=1=a^2, asa=s^{-1}\rangle$$

Let $N=\langle s\rangle=\{s, s^2, \dots, s^{n-1}, s^n=1\}$.
We have that $[D_n:N]=\frac{|D_n|}{|N|}=\frac{2n}{n}=2$.
So $N$ is a normal subgroup of $D_n$. Therefore, each subgroup of $N$ is a normal subgroup of $D_n$. (As shown in the thread http://mathhelpboards.com/linear-abstract-algebra-14/normal-subgroup-g-17852.html )

Let $T\neq N$, be a normal subgroup of $D_n$.

So, $\exists \ as^k$, for some $k=1, 2, \dots , n$ with $$s^{-i}(as^k)s^i\in T , \ \forall i=1, 2, \dots , n$$

Since $asa=s^{-1} \Rightarrow as=s^{-1}a \Rightarrow s^{-i}a=s^{-i+1}(s^{-1}a)=s^{-i+1}as=s^{-i+2}(s^{-1}a)s=s^{-i+2}as^2=\dots =as^i$, we have that $$s^{-i}as^ks^i\in T , \ \forall i=1, 2, \dots , n \\ \Rightarrow as^is^ks^i\in T , \ \forall i=1, 2, \dots , n \\ \Rightarrow as^{k+2i}\in T , \ \forall i=1, 2, \dots , n$$

We also have that $$a(as^k)a\in T \Rightarrow s^ka \in T\Rightarrow as^{-k}\in T$$

Then the product of these is also an element of $T$, $as^{-k}as^{k+2i}\in T \Rightarrow aas^ks^{k+2i}=s^{2k+2i}=(s^2)^{k+i}\in T$.

Now we have to take cases if $n$ is odd or even, or not? (Wondering)

If $k$ is even, say $k=2j$, we have that $as^{k+2i}\in T \Rightarrow as^{2j+2i}\in T\Rightarrow a(s^2)^{j+i}\in T\Rightarrow a\in T$.
If $k$ is odd, say $k=2j+1$, we have that $as^{k+2i}\in T \Rightarrow as^{2j+1+2i}\in T\Rightarrow as(s^2)^{j+i}\in T\Rightarrow as\in T$.
Does these two cases hold in both cases, when $n$ is odd and even? (Wondering)
 
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FAQ: How can we find all the normal subgroups?

What is a normal subgroup?

A normal subgroup is a subgroup of a given group that is invariant under conjugation by elements of the larger group. In other words, if the group is G and the subgroup is H, then for any element g in G and any element h in H, the element ghg^-1 is also in H.

Why is it important to find all the normal subgroups of a given group?

Finding all the normal subgroups of a group is important because it allows us to better understand the structure of the group and its properties. It also helps in simplifying computations and proofs involving the group.

How can we determine if a subgroup is normal?

A subgroup can be determined to be normal by checking if it satisfies the definition of a normal subgroup. This means that for every element in the subgroup, its conjugates by elements from the larger group must also be in the subgroup.

Are there any methods or algorithms for finding all the normal subgroups of a given group?

Yes, there are various methods and algorithms for finding all the normal subgroups of a group. These methods can vary depending on the type of group and its properties. Some common methods include the use of group generators, coset enumeration, and character theory.

Can all groups have a finite number of normal subgroups?

No, not all groups have a finite number of normal subgroups. For example, the infinite group of integers has infinitely many normal subgroups. However, many finite groups can have a finite number of normal subgroups, and in fact, this number can provide important information about the group.

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