Questions tagged [dihedral-groups]

For questions on dihedral groups, the group of symmetries of a regular polygon, including both rotations and reflections

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15
votes
1answer
674 views

Group cohomology of dihedral groups

If $m$ is odd, the group cohomology of the dihedral group $D_m$ of order $2m$ is given by $$H^n(D_m;\mathbb{Z}) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/(2m) & n \equiv 0 \bmod 4, ~ n &...
14
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3answers
8k views

Proof that $S_3$ isomorphic to $D_3$

So I'm asked to prove that $$S_{3}\cong D_{3}$$ where $D_3$ is the dihedral group of order $6$. I know how to exhibit the isomorphism and verify every one of the $6^{2}$ pairs, but that seems so long ...
14
votes
2answers
15k views

Subgroups of $D_4$

I need to determine the subgroups of the dihedral group of order 4, $D_4$. I know that the elements of $D_4$ are $\{1,r,r^2,r^3, s,rs,r^2s,r^3s\}$ But I don't understand how to get the subgroups..
11
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1answer
452 views

The smallest symmetric group $S_m$ into which a given dihedral group $D_{2n}$ embeds

Several questions, both here and on MathOverflow, address the issue of determining for a given group $G$ the smallest integer $\mu(G)$ for which there is an embedding (injective homomorphism) $G \...
10
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0answers
135 views

Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
9
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1answer
999 views

Finding Sylow 2-subgroups of the dihedral group $D_n$

I am trying to describe the Sylow $2$-subgroups of an arbitrary dihedral group $D_n$ of order $2n$. In the case that $n$ is odd, $2$ is the highest power dividing $2n$, so that all Sylow $2$-...
8
votes
2answers
251 views

Confusion about centraliser of $D_5$ in $GL_4(\mathbb{Z})$

I am trying to follow a derivation on a very old paper. My knowledge of group theory is limited, I have the basis but not much experience with advanced concepts. We are working in 4 dimensions, so ...
8
votes
1answer
1k views

Field extension with dihedral Galois group

In an old exam of my Galois Theory class there is the following question which troubles me: Let $p \neq 2$ be a prime number and $k \geq 1$ an integer. Give an example of a galois extension $L/K$ ...
7
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4answers
1k views

Some Subgroup of Dihedral Group is Normal

I ran into this question when I was studying for my abstract algebra midterm. Show that the subgroup $H$ of rotations is normal in the dihedral group $D_n$. Find the quotient group $D_n/H$. I'm ...
7
votes
1answer
5k views

Center of dihedral group

I am trying to solve the following exercise about the dihedral group and its center: If $g\in Z(D_{2n})\Leftrightarrow ga=ag, bg=gb$, where $a,b$ are generators of $D_{2n}$. We have defined the ...
6
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5answers
11k views

Is it true that a dihedral group is nonabelian?

Is it true that a dihedral group is nonabelian? I'm not sure if the result is true. I checked it for some lower order and I think the result may correct. But I failed to prove/disprove the result.
6
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2answers
3k views

How to determine what group a Galois group is isomorphic to

Consider $x^{4}-2=(x+\sqrt[4]{2})(x-\sqrt[4]{2})(x+i\sqrt[4]{2})(x-i\sqrt[4]{2}) \in \mathbb{Q}[x]$. Let $K=\mathbb{Q}(\sqrt[4]{2},i)$ be the splitting field of $x^{4}-2$. Since $K$ is a splitting ...
6
votes
1answer
2k views

Prove that the dihedral group $D_4$ can not be written as a direct product of two groups

I like to know why the dihedral group $D_4$ can't be written as a direct product of two groups. It is a school assignment that I've been trying to solve all day and now I'm more confused then ever, ...
6
votes
2answers
41 views

Finding $N(D_{4})/D_{4}$ for $D_{4}$ in $D_{16}$

I want to find $N(D_{4})/D_{4}$ where $N(D_{4})$ is the normalizer of $D_{4}$ in $D_{16}$. I'm not too clear on what the normalizer of $D_{4}$ in $D_{16}$ Is there a nice way to find $N(D_{4})/D_{4}$?...
5
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3answers
1k views

On the centres of the dihedral groups

In an proof that I recently read, the following 'fact' is used, where $D_{2n}$ denotes the dihedral group of order $2n$: If $n$ is even, then $D_{2n} \cong C_2 \times D_n$. The (short) given ...
5
votes
1answer
795 views

Dihedral group as a matrix group

I wish to consider the dihedral group as a matrix group. One way to do that is to consider it as a finite subgroup of $O_2$, a group of orthogonal $2\times 2$ matrices, defined by $\{R_0,R_1,R_2,\...
5
votes
1answer
1k views

Subgroup structure of dihedral groups

I am currently looking into structure of dihedral groups; I am interested in their subgroup structure. Dihedral group have two kinds of elements; I will use their geometric meaning and call them ...
5
votes
3answers
210 views

How does one enumerate $\mathrm{Aut}(G)$, or at least compute $|\mathrm{Aut}(G)|$?

I know that the automorphisms in $\mathrm{Aut}(G)$ preserve the order of elements of $G$, so if $G$ is partitioned according to $\mathrm{Ord}$ (order), the product of the cardinalities of the ...
5
votes
1answer
423 views

Could someone explain chirality from a group theory point of view?

While answering this question my interest in the rotation/reflection group was piqued. I personally know very basic group theory, not much more than what a group really is. I understand that the ...
5
votes
1answer
87 views

Prove order of a group is even

I am trying to solve this question and wanted to know whether my proof was correct. Suppose that $n \geq 3$, $n$ is odd, $G$ is a non-trivial group and $\varphi : D_{2n} \rightarrow G$ is a ...
4
votes
5answers
250 views

Fastest way to show that $D_6 \to S_5$ is an injective homomorphism

I want to show that there is an injective homomorphism from $D_6 \to S_5$ where $D_6$ denotes the dihidral group of order 12 and $S_5$ the symmetric group. But I'm not sure how I can do this ...
4
votes
3answers
2k views

$\mathrm{Aut}(D_4)$ is isomorphic to $D_4$

Problem statement: I need to find out if $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$ and explain my answer. I already know that it is isomorphic, so now all I need to do is to prove it. I assume that ...
4
votes
2answers
281 views

Center of $D_6$ is $\mathbb{Z}_2$

The center of $D_6$ is isomorphic to $\mathbb{Z}_2$. I have that $$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$ $$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$ My ...
4
votes
2answers
126 views

Is this Cayley table correctly computed? If so, is it correct way of presentation of this dihedral group?

Is the following table for $D_4$ correct? $$\begin{array}{c|c|c|c|c|c|c|c|c|} * & 1 & g & g^2& g^3& f& fg & fg^2 & fg^3 \\ \hline 1 & 1 & g & g^2 &...
4
votes
2answers
96 views

The number of groups $G$ (up to isomorphism) such that $G/\mathbb{Z}_3\cong D_{2n}$

I am trying to find the number of groups $G$ (up to isomorphism) such that $G/\mathbb{Z}_3\cong D_{2n}$, where $\mathbb{Z}_3$ denotes the cyclic group of order $3$ and $D_{2n}$ denotes the dihedral ...
4
votes
1answer
326 views

Why is this group of matrices isomorphic to the dihedral group?

I am reading Abstract Algebra, Theory and Applications by Judson and in exercise $13$ chapter $9$, Isomorphisms, I need to prove that the set of matrices $$A=\pmatrix{ \omega & 0 \\ 0 & \omega ...
4
votes
1answer
1k views

Composition Series for Dihedral Groups

I have been thinking about a composition series for $D_{14}\times D_{10}$ (where $D_{2n}$ is the dihedral group with $2n$ elements). Is the following a correct composition series for $D_{10}\times D_{...
4
votes
2answers
122 views

Showing that $H\leq S_n$ containing rotations must be isomorphic to $D_n$

Let $H$ be a subgroup of $S_n$ such that $H$ is isomorphic to the dihedral group $D_n$. Let also $K$ be a subgroup of $S_n$ such that $K$ is isomorphic of $D_n$. I would like to show that if $K$ ...
4
votes
2answers
91 views

What is Gal$_\mathbb{Q}(x^4 + 5x^3 + 10x + 5)$?

Find Gal$_\mathbb{Q}(x^4 + 5x^3 + 10x + 5)$ and Gal$_\mathbb{Q}(x^4 - 2)$ I was trying the second one which I think is the easiest case. However I am not able to prove it. Here is what I know (1) ...
4
votes
1answer
67 views

Does $D_4$ have a verbal subgroup of order 4?

Does $D_4$ have a verbal subgroup of order 4? How did this question arise: In the comments $Q_8$ ad $D_4$ were pointed to be a possible counterexample to this question: Is it true, that for any two ...
4
votes
1answer
114 views

How to convert one presentation into another? Please explain using a dihedral group as an example.

How can we convert a given presentation of a group $G$ into an another presentation? Would anyone please explain to me by converting two different presentations of a dihedral group? Thanks in ...
4
votes
1answer
1k views

subgroups of the group of pentagon symmetries

The pentagon has 5 line symmetries and therefore we will have 10 symmetries. So, we let the group G with order 10 denote the symmetry group of a pentagon. A subset $H$ of $G$ is a subgroup $(H, *)$ ...
4
votes
0answers
54 views

Principled way to find a shape with symmetries given by a group

Recently I've learned how groups correspond to symmetries of objects so I've been trying to find shapes corresponding to groups that I know (all with finite groups). For example, I know $\mathbb Z / ...
4
votes
0answers
175 views

Homology groups of the Dihedral Group.

I'm looking for some references for the computation of the homology in integer coefficients of the dihedral group $D_n$. Most precisely, I am interested in explicit pairing relations with the well-...
4
votes
0answers
990 views

In how many different ways can one color the vertices of a regular pentagon into four colors?

I am trying to find the number of ways to color a pentagon with 4 colors up to symmetries. I know that I should be using Burnside's Theorem, and so far I know that the group $D_5$ should act on the ...
3
votes
3answers
231 views

Why are automorphisms of $D_{2n}, n \geq 5$ odd, not always inner?

The dihedral group of order $2n$ is often presented as $$D_{2n}= \langle r,s: r^n=s^2=1, rs= sr^{-1} \rangle \text{,}$$ where $r$ denote rotations and $s$ denote reflections of a regular $n$-gon. ...
3
votes
1answer
732 views

Is a group defined by its generator set and relations?

I'm learning about generators from Dummit and Foote. They call this a presentation of the dihedral group: $$D_{2n} = \left< r,s\,|\, r^n=s^2=1,\, rs=sr^{-1}\right>$$ Does this type of "...
3
votes
3answers
156 views

Can the dihedral groups be seen as subgroups of each other?

I am solving one difficult problem and now I need information on this: If $m\mid n$, is then possible to "embed" $D_m$ to $D_n$, or otherwise said, does $D_n$ have a subgroup isomorphic to $D_m$? ...
3
votes
4answers
65 views

Elements of order 2 in $D_{2n}$

Im new at this abstract algebra stuff and im not comfortable with the proofs techniques yet, so I have a question related to the elements of order $2$ in $D_{2n}$. Problem: Prove that $\{x\in D_{2n}...
3
votes
3answers
56 views

Let $n \ge 3$. Prove that there are $n!$ different one-to-one homomorphisms from $D_n$ to $S_n$

Let $n \ge 3$. Prove that there are $n!$ different one-to-one homomorphisms from $D_n$ to $S_n$. I know there are $n!$ elements in $S_n$, but this fact didn't get me anywhere. I tried many things ...
3
votes
1answer
408 views

Prove: $D_{8n} \not\cong D_{4n} \times Z_2$.

Prove $D_{8n} \not\cong D_{4n} \times Z_2$. My trial: I tried to show that $D_{16}$ is not isomorphic to $D_8 \times Z_2$ by making a contradiction as follows: Suppose $D_{4n}$ is isomorphic to $...
3
votes
2answers
636 views

Dihedral groups: Relationship between symmetries and rigid motions

In Dummit & Foote, $D_{2n},\ n \geqslant 3$ is the set of symmetries of a regular $n$-gon, where a symmetry is a rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-gon, ...
3
votes
3answers
277 views

Why Composition and Dihedral Group have reverse order of operation?

NOTE - I didn't receive any answer in here and I think because my first post is not clear, so I entirely made another example: $K={\{id,r^2,r^4,s,r^2s,r^4s}\}$ is a proper subgroup of the dihedral ...
3
votes
2answers
239 views

Prove that the lattice graph of $D_{16}$ is not planar

How do we prove that the lattice graph of $D_{16}$ is non-planar? I wanted to prove it using Kuratwoski's Theorem but was unable to do it. And to add one more question, are there any interesting ...
3
votes
2answers
95 views

Conrad's $\mathit{Dihedral\ groups}$: Rigid motions taking a regular $n$-gon back to itself carry vertices to vertices

I have been reading Keith Conrad's expository paper Dihedral groups I and I have two questions about Theorem $2.2$, which deals with the size of $D_n$. In the first part of the proof you can read ...
3
votes
2answers
46 views

Find all characteristic subgroups of the Dihedral group $D_{12}$.

In my notation $$D_{12}=\langle \rho,\tau : \rho^6=\tau^2=1,\ \rho \tau \rho=\tau\rangle$$ So firstly, I know that all characteristic subgroups are normal. Thus, the possible candidates of $D_{2n}$ ...
3
votes
1answer
84 views

Non cyclic group of order $8$ having exactly one element of order $2$

Let $G$ be a non-cyclic group of order $8$ having exactly one element of order $2$. Prove that $G$ is generated by elements $a$ and $b$ subject to the relations $a^4=1$ and $a^2=b^2$. I can start ...
3
votes
3answers
1k views

Normalizer of a Sylow 2-subgroups of dihedral groups

I can't solve the following exercise which is the last exercise in page 146 of Dummi & Foote's Abstract Algebra: Let $2n=2^ak$ where $k$ is odd. Prove that the number of Sylow 2-subgroups of $...
3
votes
2answers
76 views

Dihedral subgroups of $S_4$

Prove that in $S_4$ there are $3$ groups that are isomorphic to $D_4$. I know that the $2$-sylows of $S_4$ should be subgroups of order $8$, but to prove it is a bit tricky for me Any help would be ...
3
votes
1answer
49 views

Unique factorization of dihedral group

My goal is to prove the following about the dihedral group $D_{2n}$: Prove that every element in $D_{2n}$ has a unique factorization of the form $a^{i}b^{j}$, where $0 \leq i < n$ and $j=0$ or $...