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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Can we build a D4 matrix representation with ${\mathbb Z_2}^{4\times 4}$ matrices?

If we look at the Dihedral 4 group. There exists a trivial matrix representation that also makes it very easy to define group action on vectors: simply use 2x2 rotation matrices for the cyclic part ...
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1answer
700 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 &...
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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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1answer
177 views

How to prove isomorphism with the Dihedral group

I have a group that I'm trying to prove is isomorphic to the Dihedral group. I know that it is finite, that it is generated by two elements $\alpha$ and $\beta$ such that: $\alpha^2=\beta^n=1$ and ...
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2answers
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Let K be a non-abelian subgroup of dihedral group and $H = K \cap \{1, a, a^2, \ldots, a^{n-1}\}$ then the index of H in K is 2.

Suppose $D_n$ is the dihedral group of order $2n$ ($D_n = \{1, a, a^2, \ldots, a^{n-1},b,ab,\ldots, a^{n-1}b\}$ with $a^n = 1, b^2 = 1$ and $ba = a^{-1}b$). I have shown that if $H \leq \{1, a,\...
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How many homomorphisms are there from $D_5$ to $V_4$?

Question: How many homomorphisms are there from $D_5$ to $V_4$, where $D_5$ is the dihedral group of order $10$ and $V_4$ the Klein four-group? I've used the fact that since $V_4$ is abelian, the ...
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1answer
17 views

Find the centralizer of each element of $ D_4$

Consider the dihedral group $D_4$. Find the centralizer of each element of $D_4$. The elements of $D_4$ are $\{1,r,r^2,r^3,s,sr,sr^2,sr^3\}$ We know that $Z(D_4)=\{1,r^2\}$. Now centralizer of $r,...
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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
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understanding the commutator of dihedral group [duplicate]

Let $G=D_{2n}=⟨x,y|x^2=y^n=e, $ $yx=xy^{n-1}⟩$ I need to find $G'$ [ the commutator of G] now I understand that $G'$ is the subgroup generated from $ U=xyx^{-1}y^{-1} , $ $ \ \forall x,y \in G$ So,...
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1answer
22 views

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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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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1answer
44 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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0answers
21 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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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
60 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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2answers
102 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 ...
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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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34 views

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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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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1answer
32 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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1answer
95 views

Classification of the irreducible group representations of the dihedral groups

Let $D_n$ be the dihedral group of order $2n$. Show that all irreducible representations have vector space dimension $1$ or $2$, and describe them up to isomorphism. Any hints how to even start?
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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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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
30 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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2answers
71 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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1answer
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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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3answers
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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 $...
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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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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
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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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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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1answer
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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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0answers
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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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2answers
62 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
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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2answers
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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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1answer
47 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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1answer
106 views

Non-orthogonal matrix representation of dihedral groups

I found this table on here: $a$ is the first element of a dihedral group $D_n$, i.e. the rotation by an angle $2\pi/n$. Few questions on that: 1) Non real, non orthogonal representation matrices. ...
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2answers
246 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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0answers
154 views

Subgroups of the dihedral group $D_n$ modulo $Aut(D_n)$

This question is related to this math.se question. Consider the dihedral group $D_n = \langle r,s \rangle.$ Two subgroups $G, H \leq D_n$ are said to be ''isomorphic'' if there is an $f \in \rm{...
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1answer
62 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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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
38 views

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
50 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
85 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 ...
2
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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 ...
3
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1answer
51 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 $...