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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Why a dihedral group isn't a normal subgroup of a permutation group?

A claim says that two permutations are conjugate when they have the same type. I don't know how to show (using this claim) that a dihedral group $D_{2n}$ isn't a normal subgroup of a permutation group ...
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Dihedral group $D_2$

Can a dihedral group $D_2$ be a set of transformations of a line segment? If it can then what are the reflections? No matter how I think about reflecting it I always get either the identity or the ...
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A question about dihedral group

The presentation of the dihedral group is $$ D_{2n}= \langle r,s \mid r^{n}=s^{2}=1, (sr)^2=1 \rangle. $$ Now, let $G$ be a group of order $2n$. Is it true that if $G$ contains two elements $a,b$ such ...
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Dihedral group for arbitrary polygon

Is a dihedral group only considered for shapes that when reflected or rotated fit exactly back into place of the original image? My confusion arises from this wikipedia article and specifically from ...
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Inverse element of dihedral group… [closed]

I am having a hard time conceptualizing what is the inverse element of, for example, a dihedral group on the equilateral triangle. Can anyone explain this to me?
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Dihedral group understanding

I have just started reading Abstract Algebra by Dummit and Foote and I have been stuck on their explanation of dihedral groups. On page 23 Dihedral Groups there are 3 paragraphs(one paragraph is at ...
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1answer
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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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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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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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1answer
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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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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
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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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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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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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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
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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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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
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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
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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
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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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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 “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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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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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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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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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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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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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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1answer
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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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1answer
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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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2answers
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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 $H\lhd D_{2n}$? [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
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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
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
247 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
52 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
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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 ...