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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Find $|{\rm Aut}(D_8\times S_3)|$ using a particular result about Remak decompositions.

This is Exercise 3.3.9 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. The Details: On page 6, ibid, the dihedral group $D_{2n}$ ...
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34 views

How many elements of order $3$ are in $D_{12}$? [closed]

How many elements of order $3$ are in $D_{12}$? Okay so I know that only rotations can be order $3$ in $D_{12}$ because all reflections have order 2. These create a cyclic subgroup so there are $4-1=...
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1answer
33 views

A group isomorphism involving dihedral group

Show that $$\begin{align} \langle c, d: c^n , d^2, (cd)^2\rangle &\to D_n,\\ c&\mapsto r,\\ d&\mapsto s\end{align}$$ is an isomorphism, where $D_n$ is the dihedral group, $r$ represents ...
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2answers
45 views

The abstract description of the $n$th dihedral group $\mathcal{D}_n$.

Every element of $\mathcal D_n$ is of the form $r^is^j$ for $\,0\le i<n,\,0\le s<2\,$, where $\mathcal D_n=\langle\{r,s\}\rangle$ with $|r|=n,|s|=2$ and $sr=r^{n-1}s.$ I want to prove the above ...
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4answers
85 views

Prove $D_{12}$ and $S_4$ are not isomorphic groups in 3 different ways [duplicate]

Give three reasons why $D_{12}$ and $S_{4}$ are not isomorphic. More precisely, prove that $D_{12}$ and every group isomorphic to it satisfy three properties that $S_{4}$ does not satisfy. So I have ...
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2answers
69 views

Prove that $D_{12}$ is not cyclic.

So I am trying to prove that $D_{12}$ and $S_4$ are not isomorphic by giving three properties that all isomorphisms to $D_{12}$ has that $S_4$ does not. I have already proved that there is an element ...
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0answers
17 views

Proving $r^{k}$ is the only nonidentity element of $D_{2n}$ that commutes with all elements for even $n$ [duplicate]

I am working on proving that $z = r^{k}$ is the only nonidentity element of $D_{2n}$ ($n = 2k$) which commutes with all elements of $D_{2n}$. I have been able to show that any other power of $r$ ...
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1answer
31 views

Number of Sylow $2$-subgroups of $D_{2n}$

The question is from Dummit and Foote pg146, Exercise 12. Let $2n=2^ak$ where $k$ is odd. Prove that the number of Sylow $2$-subgroups of $D_{2n}$ is $k$. This is my attempt. My Attempt: The order ...
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1answer
38 views

Is $H$ is a subgroup of $D_4?$ Yes/No [duplicate]

Is $H=\{ x\in D_{4} \mid x^2=1\}$ is a subgroup of $D_4?$ My attempt : I think not Take the elements $s$ and $rs$ of $D_4$ Here $s^2=1$ and $( rs)^2 =rsr^{-1}s=s^2=1$ But $s(rs)=rs^2=r \neq 1$ ...
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1answer
29 views

Prove that there are exactly 2n symmetries of a regular sided polygon, $P_n$ .

I'm going through my lecture notes and didn't really understand a section of the proof for showing there are exactly $2n$ symmetries of a regular sided polygon. The proof states: It suffices to show ...
2
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1answer
40 views

For $n,i>0$ find a normalizer of $\langle \tau \sigma^i\rangle$ in $D_{2n }$

For $n,i>0$ find a normalizer of $\langle \tau \sigma^i\rangle$ in $D_{2n }$. I would like to know if my reasoning holds and I also block at one point. If someone could give a feedback and help (...
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1answer
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Dihedral group presentation

I am solving the following question from Aluffi. I am stuck on showing that this is the actual presentation of $D_{2n}$. Question: Describe generators and relations for dihedral groups $D_{2n}$. My ...
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Twisted subgroups. (Subgroups of $D_5 \times S^1$

Heya all I am looking for twisted subgroups of $D_5$, equivalently subgroups of $D_5 \times S^1$, and was wondering if the group generated by the elements $[\rho,0]$ and $[\kappa,\frac{1}{2}]$ forms ...
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0answers
41 views

Can I make a lattice of the classes of subgroups related by outer automorphism for D8xC2?

While trying to come up with a coloring scheme to organize all the complex interactions of the D8×C2 lattice, I realized I can make a much simpler lattice like structure of the kind of subgroups ...
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0answers
21 views

Composition of rotation and reflexion of two symmetries of a regular $n$-gon.

I know that composition of a rotation and reflexion of a regular $n-$gon in plane is reflexion and can be proved by using rotation and reflexion matrices . But I want to prove it by the fixed point ...
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20 views

Prove that $D_n$ is generated by ⟨r,f⟩ [duplicate]

I want to prove the dihedral group is generated by a rotation r and a reflection f (flip). I managed to prove this relation $fr = r^{n - 1}f$   1.0 and in addition there are the simple relations $f^2$ ...
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2answers
57 views

Presentations of the dihedral and quaternion groups

I am reading Lang’s algebra text and he is currently discussion generators of groups and gives the following examples: Is Lang suggesting here that any two elements of either group which satisfy ...
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0answers
27 views

Is $D_{2^{n+1}}/Z_2\cong D_{2^n}$? [duplicate]

I run into a problem when I try to solve a question, show $D_{2^{n+1}}/Z(D_{2^{n+1}})\cong D_{2^n}$. Note: $D_n$ is the dihedral group of order 2n. I have shown $D_{2^{n+1}}/Z(D_{2^{n+1}})\cong D_{2^{...
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32 views

Ascending central series for $D_8$

My Group Theory notes give as definition for an ascending series: $G_0, G_1,...,G_n \unlhd G $ such that $\{1\} = G_0 \unlhd G_1 \unlhd ... \unlhd G_n=G$ and $G_{i+1}/ G_{i} \subset Z(G/G_{i})$. As an ...
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25 views

Importance of studying certain graphs on the dihedral group

I am actually not sure if this is the right platform to ask this, but I hope someone can enlighten me. I am an undergraduate math student, and I recently started reading academic papers on graph ...
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1answer
91 views

Do overlapping left and right coset spaces have a name?

I have been studying they symmetries of the square, D₄. I would like to use the subgroups that are not normal to make a coset space, but since they are not normal, the right coset space is different ...
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1answer
32 views

Tracial states on $C_\text{r}^\ast(\mathbf{D}_\infty)$, $\mathbf{D}_\infty$ being the infinite dihedral group

Let $\mathbf{D}_\infty$ be the infinite dihedral group, i.e. the group generated by two elements $s$ and $t$ with $s^2=t^2 =e$ which are free with respect to each other. Consider the reduced group $C^\...
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Subgroups of dihedral group $D_4$

I'm trying to find all of the subgroups of the dihedral group, $D_4$, of the square. I will exhibit the group as follows. Let $r$ be a counterclockwise reflection through $\frac{\pi}{2}$ radians, ...
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0answers
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Is the generalized quaternion group $Q_{n}$ isomorphic to the dihedral group $D_{2^{n-1}}$?

this is an exercise in the book "Elementos de Álgebra", a brazillian book of Algebra by Arnaldo Garcia and Yves Lequain (This is not a homework, I'm doing this by myself). Let $n \geq 3$ an ...
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1answer
99 views

Proof that the order of $D_n$ is $2n$

Prove that the order of $D_n$ is $2n$. I'm trying to find a proof for this, but couldn't find anything on the site. I found this, but it's not exactly what I'm looking for. They're mentioning ...
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0answers
31 views

Group Presentation of the Dihedral Group $D_{2n}$ [duplicate]

I am working through Dummit's "Abstract Algebra" and an exercise is to show that the group $\langle x_1,y_1\vert x_1^2=y_1^2=(x_1y_1)^2=1\rangle$ is the dihedral group $D_4$. The hint is to ...
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3answers
95 views

Example of presentation of a group.

I’m reading up on presentations of a group. In here, the example is the dihedral group $D_8$, with generators rotation $r$ of order $8$, and flip $f$ of order $2$. In the construction of the ...
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74 views

How to show that the $f$ is one-one and onto?

Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $n = |xy|.$ My attempt : Since every element of $G$ can be ...
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1answer
48 views

Multiplication/Cayley tables for the Dihedral Groups

I am currently doing a group theory problem, which asks for the multiplication table of the dihedral group $D_4$. Having looked up the answer online, I do not understand how some of the elements arose....
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74 views

Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order 2 in $G$ that generate $G$. Prove that $G\cong D_{2n}$, where $n=|xy|.$

Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order 2 in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $n = |xy|.$ My attempt : This question already asked here....
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1answer
112 views

$D_{2n}$ is isomorphic to a subgroup of $S_n$ (for $n>2$).

All the proofs I've come across of the fact in the title call into play the action of the group on the vertices of the regular $n$-gon, i.e. they rely on the geometrical definition of $D_{2n}$ (in ...
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1answer
27 views

Sylow 2-subgroup and Sylow 3-subgroup of $D_{24}$ (of order $48$).

Find Sylow 2-subgroup and Sylow 3-subgroup of $D_{24}$ I have found $n_2 = 1$ or $3$ and $n_3 = 1$ or $4$ or $16$, where $n_2$ & $n_3$ are the number of Sylow $2$-subgroups and Sylow $3$-...
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2answers
111 views

Are $D_{20}$ and $\mathbb{Z}_{2} \times \mathbb{Z}_{10}$ isomorphic?

Are $D_{20}$ and $\mathbb{Z}_{2} \times \mathbb{Z}_{10}$ isomorphic? For $D_{20}$ I am using the notation $D_{2n}$ and $\mathbb{Z}_{2}, \mathbb{Z}_{10}$ are additive groups. I think not, and my ...
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0answers
32 views

In $S_4$ describe all the leftcosets of the subgroup $D_4$

How can I describe the left and right cosets of $D_4$ I know that $D_4 $ is $ \begin{array}{ll} \rho_{0}=\left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{array}\...
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1answer
123 views

Why is $A \rtimes B \simeq \mathbb Z \rtimes \mathbb Z/2\mathbb Z$

Hello everyone I have a hard time trying to resolve this problem if anyone could help it would be a lot appreciated. Let $$f_1\colon\mathbb R\rightarrow \mathbb R,\,f_1(x)=-x,\quad f_2\colon\mathbb R\...
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0answers
48 views

Find all subgroups of the dihedral group $D_n$

For $n\in \mathbb{Z}^+$ and $k < n,$ let $R_k = \begin{bmatrix}\cos (\theta_k) & -\sin(\theta_k)\\ \sin(\theta_k) & \cos(\theta_k)\end{bmatrix}$ and $F_k = \begin{bmatrix}\cos (\theta_k) &...
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0answers
37 views

Reference for theorem on quotient of generalized quaternion group by its centre being isomorphic to dihedral group

I am looking for a book/paper reference for the following theorem; Suppose $ n \geq 3 $. Let $ Q_{4n} $ be the generalized quaternion group of order $ 4n $. Then $ | Z(Q_{4n} ) | = 2 $ and $ Q_{4n} / ...
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0answers
66 views

Orbits of dihedral group

How can I determine orbits of dihedral group $D_n$? Any relevant references will be useful. I consider cycle graph with $n$ vertices and (as I understand) its automorphism group is $D_n$. Then, ...
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1answer
52 views

Symmetries in a dihedral group

How many symmetries does $D_{2n}$ have exactly? I imagine it has rotations written in the form $\frac{x\pi}{n}$ such that $x \in \{0, 1, \dots, n-1\}$, and can also be flipped. But what about ...
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1answer
26 views

Find subgroup(s) of order 2 of $D_{2p}$

Suppose that $p$ is a prime greater than 2. Dihedral group $$ D_{2p}=\left\{ 1,a,a^2,\cdots ,a^{p-1},b,ab,a^2b,\cdots ,a^{p-1}b|ord\left( a \right) =p,ord\left( b \right) =2,ab=ba^{p-1} \right\} . $$ ...
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1answer
32 views

For each positive integer $n \geq 3$, determine the centre of $D_{2n}$ (help for understand) [duplicate]

For each positive integer $n \geq 3$, determine the centre of $D_{2n}$. Proof. Let $F$ be any flip and $R$ any rotation. Drawing out the effects of each operation, we find that $FR = R^{−1}F$. This is ...
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26 views

Composition series of $D_{12}$ (of order $12$).

I find that $$\{id,\langle r^2\rangle, \langle r\rangle,D_{12}\}$$ and $$\{id,\langle r^3\rangle, \langle r\rangle,D_{12}\}$$ are composition series. Are there any more composition series of $D_{12}$? ...
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1answer
36 views

Finding a normal subgroup of a fiber product giving a prescribed quotient.

Let $D_{2^{n-1}}$ be the Dihedral group of order $2^n$. Letting $s_1:D_{2^{n-1}}\rightarrow \mathbb{Z}_2$ be determined by one of its subgroups isomorphic to $D_{2^{n-2}}$. The book I'm reading says ...
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1answer
43 views

To determine the cardinality of the rotation group of a cube

In Dummit and Foote's Abstract algebra a problem is given as- Let $G$ be the group of rigid motions in $\mathbb R^3$ of a cube. Show that $|G|$=24. For an n-gon, all the points on the n-gon can be ...
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1answer
42 views

Why does rotation $r^2$ by $\pi$ give an order $2$ subgroup in $D_8?$

Find a subgroup of $D_8$ of order $2.$ My attempt : A presentation of $D_{2n}$ is $\langle r,s\mid r^n=s^2=\mathrm{id},srs=r^{-1}\rangle$, the reflections are $s,sr,sr^2,\ldots,sr^{n-1}$ Put $ n=4$ ...
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2answers
60 views

For what $m,n$ is $D_{mn}$ isomorphic to $D_m \times Z_n$?

For what $m,n$ is $D_{mn}$ isomorphic to $D_m \times Z_n$? Should I try to define an isomorphism between them? Where to start?
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1answer
34 views

Prove that any dihedral group can be generated by symmetry and reflections.

One definition of a dihedral group $D_n$ is $$\langle s,r : s^2=r^2=(sr)^n=1 \rangle.$$ where $s$ is a symmetry along an axis and $r$ is a rotation of $2\pi/n$. I was just thinking this is how ...
0
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1answer
28 views

Prove that $D'_n=\langle x^2\rangle$

I want to check if my solution to one problem from my group theory course is valid. The problem is: Given $D_n=\{x^iy^j:0\leq i<n,0\leq j<2\}$, prove that $D'_n=\langle x^2\rangle$. My attempt ...
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0answers
46 views

Prove three statements about the dihedral group $D_6$

I'm trying to solve this problem from my group theory course: Consider the dihedral group $D_6$ of isometries of the euclidean plane which fix a regular hexagon: (a) Prove that this group is ...
4
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1answer
55 views

Finite group generated by two different order $2$ elements is $\cong$ to $\mathbb{Z}_2^2$ or $D_n$

I'm trying to solve this problem from my group theory course: Given $G$ finite group generated by two different order $2$ elements. Prove that $G\cong \mathbb{Z}_2^2$ or $G\cong D_n$ for some $n\geq ...

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