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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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Dihedral groups non-commutavity

Here is a result about dihedral groups. $rs = sr ^{-1}$, where $r$ is a rotation of $\frac{2 \pi}{n}$ radians and $s$ is a reflection about the line of symmetry from vertex $i$ and the origin. This is ...
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Confusion about a proof about subgroups of dihedral groups

This article shows that every subgroup of $D_n = \langle r, s \rangle$ is cyclic or dihedral. Theorem 3.1. Every subgroup of $D_n = \langle r, s \rangle$ is cyclic or dihedral. A complete listing ...
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Show the Dihedral Group $D_n$ is generated by rotations and reflection along the x axis.

I'm having problems understanding the excersice: E) Define $D_n$ as the group of symmetries of a regular n-gon. Name the vertices $V=\{V_0,V_1,...,V_{n-1}\}$ so that $$V_{k}=\exp({i\cdot\dfrac{2\pi k}{...
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Certain Isomorphic Representations of the dihedral group $D_{3}$

Using the following presentation of the dihedral group $D_{3}$ \begin{equation} D_{3} = \left\langle r,s \mid r^{2} = s^{2} = (rs)^{3} = e \right\rangle \end{equation} There is one (...
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Nonabelian dihedral groups and a question in number theory [duplicate]

I'll use a concrete definition of a dihedral group $D_{2n}$ which emphasizes its group structure: $D_{2n}$ consists of distinct elements $r_0,...,r_{n-1},s_0,...,s_{n-1}$ so that for any $i \in \...
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1answer
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Find representative for each conjugacy class of $D_{10}$

How do you find representative for each conjugacy class of $D_{10} = \langle r \rangle_5 \rtimes \langle s \rangle_2$? I know $D_{10}$ has $4$ conjugacy classes which are: $[Id]$, $[r]$, $[r^2]$, $...
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Let $D_{2n}$ be the dihedral group of order $2n$. Let $H$ be the set of rotations of the regular $n$-gon. Is $H\lhd D_{2n}$?

Let $D_{2n}$ be the dihedral group of order $2n$, i.e., the group of symmetries of the regular $n$-gon. Let $H$ be the set of rotations of the regular $n$-gon. Is $H\lhd D_{2n}$?
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Compute a set $S$ given information about how it is acted upon transitively by $D_8$

Let $D_8=D_{2 \cdot 4}$ be the dihedral group on a regular $4$-gon. Suppose that $S$ is a subset of $S_4$, such that S contains the element $( 1 \ 2 \ 3)$. We also know that $D_8$ acts transitively ...
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How to show that $|D_{2n}| = 2n$ via the presentation?

Consider the dihedral group $$D_{2n}= \langle a,b \mid a^n = 1 = b^2, b^{-1}ab = a^{-1}\rangle$$ How can I show that $|D_{2n}| = 2n$? I'm trying to show that we can write every element in the form ...
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1answer
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Non-trivial isomorphism between the dihedral group to itself.

I want to find a non-trivial isomorphism between the dihedral group $D_n$ and itself. Non-trivial means that the isomorphism won't be the identity. I looked at the group $D_n$ as the set of the ...
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Group actions of $D_5$

I have to give $5$ examples of $D_5$ acting on a set. So far, I have $D_5$ acting on the set of vertices of a pentagon and “rotating” each vertex one to the right, sending the vertices to a reflection ...
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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 $...
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A formula for the number of order $2$ elements of $D_m\times D_n$ for even $m>2$ and odd $n>2$. (Gallian 8.24.)

This is Exercise 8.24 of Gallian's "Contemporary Abstract Algebra (Eighth Edition)". Answers that use material from the textbook prior to the exercise are preferred. Presentations, for instance, are ...
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Group Theory - Dihedral Groups

Two questions related to Dihedral groups: What is the conventional notation for Dihedral groups? Is it Dn where n is the number of sides in a regular n-gon, or is it D2n where n is the number of ...
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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 ...
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1answer
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Set of Rotations Cyclic?

For the dihedral group $D_{n}$ of order $2n$, is the group $R$ formed by its $n$ rotations cyclic in general? Or is the factor group $D_{n}/R$ cyclic? I am trying to show the series $D_{n}>R>(1)$...
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Showing two series are not the same.

I want to show that the following two composition series are not the same: $D_{8}\triangleright \left \langle s,r^{2} \right \rangle \triangleright \left \langle s \right \rangle \triangleright (1)$ ...
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Coloring triangular dihedral #2

To start with, my dihedral is a bit specific, here is a picture I need to find amount of ways to color faces ( there are 8 ) into 3 colours. I have already something in my mind because of help ...
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Derived series of the Dihedral group [duplicate]

I'm working on derived subgroups because I'm studying for an exam and I want to show that in the case of the dihedral group $D_{2n}=\langle\sigma ,\tau|\sigma^n=\tau^2,\sigma^{\tau}=\sigma^{-1}\rangle$...
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Quotient group of the dihedral group by $\langle r^2 \rangle.$

Show that $G/H$ is abelian, where $G$ is the dihedral group $$ G={\langle r,\, f \mid r^n=f^2=1,\, rf=fr^{-1}\rangle}$$ and $H$ is the subgroup $\langle r^2 \rangle.$ I've tried showing that for $...
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1answer
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Coloring sides of truncated triangular dihedral(bipiramid) into 3 colours

I need to find out the amount of ways to colour truncated triangular dihedron into 3 colours. So, the task will be easier if I had simple triangular dihedron. First of all, do I understand right ...
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Find a group $G$ with $a\in G$ such that $|a|=6$ but $C_G(a)\neq C_G(a^3)$.

This is part of Exercise 46 of Chapter 3 of Gallian's "Contemporary Abstract Algebra". Notation 1: The centraliser of $g$ in a group $G$ is denoted $C_G(g)$. Notation 2: The dihedral group $...
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If $G$ has a nontrivial centre, must every subgroup of index $3$ be normal?

If a group $G$ has a nontrivial centre, must every subgroup of index $3$ be normal? $S_3$ yields an example of a group with a non-normal subgroup of index $3$, although it has a trivial centre. ...
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1answer
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Automorphism of $D_8$ [duplicate]

I am trying to prove that $Aut(D_8) \equiv D_8$. It is not hard to see that $\lvert Aut(D_8)\rvert = 8$. Indeed, it is at most $8$ as $r$ (canonical rotation) has order $4$ and $s$ (canonical ...
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What is $D_{16}/ Z(D_{16})$?

I was asked the following: Let $D_{16}$ be the dihedral group of order $16$. What is $D_{16} / Z(D_{16})$? I know that the center of $D_{16}$ har order $2$. So therefore, the quotient has order $16/2 ...
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Find $G/Z(G)$ given the following information about the group?

$G$ is a finite group generated by two elements $a$ and $b$, we are given the following data: Order of a= $2$ Order of $b=2$ Order of $ab=8$. If $Z(G)$ denotes the center then what is $G/Z(G)$ ...
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Quick question about colouring the vertices of a fixed square using three colours.

If a square remains fixed in the plane, how many different ways can the corners of the square be colored if three colors are used? Why does the answer use $D_{4}$ when the square cannot move? I don't ...
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Finding Homomorphisms from dihedral groups to cyclical groups

Ok so there was another question very similar to this on here however it leaves me a little confused. $\bf{Question}$ Let G = $D_{14}$ the Dihedral group order 14 and A = $c_7$ be the cyclical ...
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Prove the existence of homomorphism.

I am trying to answer the following question. Is there any group homomorphsim $\phi: D_4 \rightarrow S_5$?
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isomorphism of dihedral group with these elements

So I have a group of order $2m$ with these elements: $$(\overline{0},\overline{0}),(\overline{1},\overline{0})...(\overline{m-1},\overline{0})$$ $$(\overline{0},\overline{1})(\overline{-1},\overline{1}...
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Proper condition on the dihedral group

Is there a theream which is a condition on $n\in\mathbb N$ that says when the dihedral group, $D_{n}$, has non-cyclic subgroups? After spending some time figuring a condition I tried to find some ...
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1answer
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Find $ C_{D_{12}}(a)$=centraliser of $a$ and $ C_{D_{12}}(b)$=centraliser of $b$

Consider the Dihedral group $D_{12}=\left\langle a,b: a^6=e, \ b^2=e, \ ba=a^5b \right\rangle$ of order $12$ of symmetries of regular hexagon. Every element of $D_{12}$ can be written as $a^ib^j, \ 0 \...
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$G_{x_1x_2+x_3x_4}\cong D_8$

Let $R=F[x_1,x_2,x_3,x_4]$ be the set of polynoms in 4 variables over a field $F$. Let a map $\varphi:S_4\to \operatorname{Sym}(R)$ by $$ \\ (f(x_1,x_2,x_3,x_4))\varphi(\sigma)=f(x_{1\sigma},x_{2\...
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Determine number of conjugacy class in $D_8$

I’m using the formula that the number of conjugacy class is given to be $\frac{1}{|G|}\sum|C_{G}(g)|$, where $C_{G}(g)=\{h \in G ; gh=hg\}$, which is a special result by Burnside’s theorem. I found ...
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Is $G$ isomorphic to the dihedral group $D_{10}$?

Let $G\le S_6$ be the subgroup generated by the permutations $\sigma=(12356)$ and $\tau=(26)(35)$. I'm asked to determine: (a) the order of $G$ and the period of each element; (b) if $G$ is ...
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How to show $D_3\oplus D_4$ is not isomorphic to $D_{24}$? [duplicate]

How to show $D_3\oplus D_4$ is not isomorphic to $D_{24}$? Here $D_n$ is the dihedral group of order $2n$. I am not sure how to prove this. I am not very good with the dihedreal groups.
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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 ...
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1answer
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No common eigenvectors then representation irreducible

Embed an equilateral triangle into $\mathbb{R}^2$ with vertices $(1,0), (\frac{-1}{2}, \frac{\sqrt{3}}{2}), (\frac{-1}{2}, -\frac{\sqrt{3}}{2})$. Counterclockwise rotation and reflection over the $x$ ...
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1answer
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Normal subgroup generated by $D_8$

Let $D_8 = <a,b : a^4 = b^2 = 1, bab^{-1} = a^{-1}>$ be the dihedral group. I'm trying to show that the subgroup generated by $a^2$ is normal. But, isn't $<a^2> ={\{1, a^2}\}$? So the ...
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Is dihedral group defined as following commutative?

Let $G$ be the dihedral group defined as the set of all formal symbols $x^iy^j$, $i=0,1$, $j=0,1,\ldots,n-1$, where $x^2=e$, $y^n=e$, $xy=y^{-1}x$. EDIT - My proof is wrong .But i will be thankful to ...
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1answer
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Counting the elements of a dihedral group $D_n$ of a $n$-sided regular polygon

For a $n$-sided regular polygon there are $n-1$ possible rotations: $a,a^2,a^3,a^{n-1}$, a 1 reflection $b$, 1 identity $e=a^n=b^2$. There are also 2(n-1) elements $ab,a^2b,a^3b,...a^{n-1}b$ and $ba,...
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1answer
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How to prove the following facts about Dihedral Groups

I am reading about Dihedral Groups and I have following questions: Elements of $D_n$ act as linear transformations of plane. My thought:I know that $D_n=\{\langle a,b\rangle :a^n=b^2=1,bab=a^{-1}\}$ ...
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Dihedral group element properties from Dummit and Foote

I am self studying Dihedral groups from Dummit and Foote abstract algebra book. It is given to prove: $(1)$ $s\neq r^i$,$(2)~sr^i\neq sr^j,0\le i,j<n$ where $r $ is rotation by $2\pi /n$ angle ...
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1answer
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Isomorphism between the dihedral group D2 and the integer group {-1,1}X{-1,1}

I put all of the elements of $D_2$ into matrix form, giving $[\begin{matrix} 1 & 0\\ 0 & 1\end{matrix}]$ , $[\begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix}] $ , $[\begin{matrix} 1 & ...
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1answer
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The dihedral group $D_8$ isn't Hamiltonian [duplicate]

Let $D_8=\{a^ib^j:i\in\{0,1\},j\in\{0,...,3\}\} $ be a dihedral group, where $$a=\begin{pmatrix} -1 &0 \\ 0 & 1 \end{pmatrix}\qquad\text{ and }\qquad b=\begin{pmatrix} cos\theta & -sin\...
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2answers
119 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 $K$ ...
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1answer
55 views

Deduce that the symmetry group of the dodecahedron is a subgroup of $S_5$ of order 60.

Let D be a regular dodecahedron. It is possible to inscribe a cube on the vertices of D thus: (a) Prove that one can inscribe exactly 5 such cubes inside D. (b) Deduce that any rigid motion of D (i....
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38 views

How to determine basis functions partners of $x$, $x^2$, and $yz$ in $D_6$?

I have a character table for $D_6$ and I am trying to understand how to find the partners of basis functions. I am starting with $x$, $x^2$ and $yz$. I am currently working on the $x$ function but am ...
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1answer
31 views

When should I use symmetric groups vs. dihedral groups with Burnside's theorem?

For the following problem: How many ways can the vertices of an equilateral triangle be colored using three different colors? From the answers in (https://math.stackexchange.com/users/128316/...
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1answer
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In dihedral group $FR=R^{-1}F$

Let F be any reflection (flip ) about axis of symmetry . And R be rotation by $\frac{2\pi}{n} $ radian counterclockwise .(n is the number of vertex). Then $FR=R^{-1}F$ I looked at some example and ...