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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For all $n > 0$ express $D_{2n}$ as a semidirect product $\mathbb Z_n \rtimes_\theta \mathbb Z_2$, finding $\theta$ explicitly.

For all $n > 0$ express $D_{2n}$ as a semidirect product $\mathbb Z_n \rtimes_\theta \mathbb Z_2$, finding $\theta$ explicitly. I am not sure how to go about finding $\theta: \mathbb Z_2 \to \...
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2answers
90 views

Find order of $X_{2n} := \langle x,y|x^n = y^2 = 1, xy = yx^2 \rangle$

Let $n$ be a multiple of $3$, i.e. $n = 3k, k \geq 1$ and consider the group $$X_{2n} := \langle x,y|x^n = y^2 = 1, xy = yx^2 \rangle $$ Show that $|X_{2n}| = 6$. (Source problem: dummit and foote, ...
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Images of the generators of $D_{10}$ under its automorphisms.

I have constructed the dihedral group generated by $a$ and $b$ of order $10$ in GAP by the following way: ...
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1answer
42 views

Dihedral group of order 10 in GAP

The dihedral group of order $10$ is given by $D_{10} = \langle a,b| a^5 = b^2 = 1, bab^{-1} = a^{-1}\rangle$. Now I need to find all the elements in GAP. But whenever I type ...
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20 views

What is the name for this generalization of dihedral groups?

A dihedral group $D$ can be defined as the group generated by elements $r$ and $s$, where $r$ has order $n$, $s$ has order $2$, and $sas = a^{-1}$ for all $a \in \langle r \rangle$. It seems that more ...
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2answers
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elements of order two in $D_{10}$

Which elements have order two in $D_{10}$? In $D_{10}$ there are $10$ elements, five of which are rotations and five reflections. Let $\rho = (1\hspace{1mm}2 \ldots 5)$ and $\tau = (1)(2\hspace{1mm}5)...
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1answer
44 views

Find all Quotient (or Factor) Groups of D4 (Dihedral Group 4)

I need to be able to find all of the quotient groups for dihedral group 4 with $D_4=${$e,R,R^2,R^3,V,H,D,D'$}. I know I have to start by finding the normal subgroups, which are {$e,R^2$} {$e,R,R^2,...
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Finding basis for a representation of $D_8$.

Let $G=D_8=\langle a,b\mid a^4=b^2=1,b^{-1}ab=a^{-1}\rangle$. The character table of $D_8$ is known and is Let $U:=\bigg\{\sum\limits_{1\leq i<j\leq 4} a_{ij}x_ix_j\mid a_{ij}\in\mathbb{C}\bigg\}$ ...
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2answers
51 views

Dihedral group generated by $\langle r,s\rangle$ for all $n$

Under wikipedia for Dihedral groups it claims the following: The $2n$ elements in $D_n$ can be written as $\{e,r,r^2,r^3,\ldots,r^{n-1},s,rs,r^2s,\ldots,r^{n-1}s\}$. I know why this is true and it ...
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Find a formula for number of orbits under action of $D_{4}$

We colour each side of a square with $k \geq 1$ colours. Find a formula for the number of orbits under the action of $D_{4}=\{ e , r,r^{2},r^{3},s,sr,sr^{2},sr^{3} \}$ on the set of colours. Now as ...
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1answer
31 views

Composition of elements in dihedral group

I've come across the following example: $$ρ^3·σρ^2 = ρ^2σρ^{−1}ρ^2 = ρ^2σρ = ρσρ^{-1}ρ = ρσ = σρ^{−1} = σρ^5$$ And was wondering if it is true in general that $ρ^i·σρ^j = σρ^{i+j}$? I know that $ρσ =...
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Two questions about the dihedral group

First question: 1) Is the sum of subgroup indices of dihedral group with $2n$ elements equal to $\sigma_2(n)+2\cdot \sigma(n)$? Second question: 2) Is $\sigma_2(n)+2\cdot \sigma(n) \le L(H(D_n))$? ...
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30 views

classification of representations of $D_{1009}$

A follow-up of this question To fix ideas, take $n=1009$. $D_n$ has $2$ irreducible representations of degree $1$ and $504$ representations of degree $2$. Are the degree 1 representations all ...
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1answer
29 views

Orders of the Elements of $D_6/Z(D_6)$

I have been trying to calculate the orders of the elements of $D_6/Z(D_6)$. For example, using $R_{60}$ to represent rotation by 60 degrees and $R_0$ to represent rotation by 0 degrees (the identity ...
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16 views

Question about conjugacy classes in dihedral groups [duplicate]

I'm trying to find the conjugacy class of a rotation $r^{k}$. Is it unitary? How about a symmetry $s$? Any ideas?
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1answer
31 views

Why is $\langle r\rangle$ characteristic in $D_n$?

I need to determinate if $\langle r\rangle$ is characteristic in $D_n = \langle r \rangle_n \rtimes \langle s \rangle_2$. This is trivial if I use the result that every cyclic group is ...
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1answer
28 views

Conjugate of elements in the Dihedral Group

I was trying to do the following from a past exam of my Rings and Groups' professor Classify all conjugacy classes of the elements in the dihedral group $D_n$ = $\{ 1,r,r^2, ... , r^{n-1} ,s ,rs ,...
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What mistake have I made in showing these automorphisms generate the group $D_8$?

In my lecture notes it says that the group generated by the automorphisms $\sigma(t)=it, \tau(t)=\tfrac{1}{t}, G=<\sigma,\tau>$ is the Dihedral 8 Group . Now $D_8=<\sigma,\tau|\sigma^4=\tau^...
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2answers
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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 ...
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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 ...
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67 views

Quotient group of dihedral group

Let $G=\{e,r^{2},...,r^{8},s,sr,...,sr^{8}\}$ and let $N=\langle r^{3} \rangle.$ Now let $\pi(g)=\bar{g}=gN$ be surjective with kernel $N$. I have to show that $G/N=\{\bar{e},\bar{r},\bar{r^{2}},\...
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2answers
40 views

How to find number of abelian subgroups of diheral group? [closed]

How to find number of abelian subgroups of diheral group $D_n $? Attempt: I have counter-examples for $n=1,2$ so I know that it isn't true for $n<3$. Is it true for $n\ge 3$? How do you know this?...
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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}...
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Understand the free group universal property applied to $D_n$

For $n ≥ 3$ and $D_n$ the dihedral group of order $2n$ with présentation $\langle r, s : r^n = s^2 = srsr = 1\rangle$ prove that for all $(a, b) \in (\Bbb Z/n\Bbb Z)^2$, there exists a morphism $f$ ...
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53 views

How can I show that $D_{2n} \cong C_n \rtimes C_2 $

Let $D_8 := \langle a,b \mid a^4 = 1 = b^2, bab = a^{-1}\rangle$ I'm trying to formally show that $$D_{8} \cong C_4 \rtimes C_2 = \langle s\rangle \rtimes \langle t \rangle$$ My book gives as hint ...
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61 views

Show that $D_{33}$ is not isomorphic to $D_{11} \oplus Z_{3}$.

Goal: Show that $D_{33}$ is not isomorphic to $D_{11} \oplus Z_{3}$. They are both non-cyclic groups of order $66$. The same orders are possible for their elements. Comparing massive Cayley Tables is ...
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1answer
38 views

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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2answers
51 views

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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1answer
35 views

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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1answer
46 views

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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2answers
27 views

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
61 views

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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1answer
47 views

Let $D_{2n}$ be the dihedral group of order $2n$. Let $H$ be the set of rotations of the regular $n$-gon. Is $H\lhd D_{2n}$? [closed]

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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1answer
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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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1answer
84 views

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
27 views

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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2answers
245 views

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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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 $...
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1answer
38 views

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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1answer
37 views

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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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 ...
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1answer
26 views

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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21 views

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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0answers
14 views

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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23 views

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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2answers
87 views

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
58 views

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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2answers
86 views

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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0answers
66 views

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
52 views

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 ...