Questions tagged [dihedral-groups]
For questions on dihedral groups, the group of symmetries of a regular polygon, including both rotations and reflections
391
questions
1
vote
1answer
21 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 \...
2
votes
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, ...
1
vote
0answers
34 views
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:
...
1
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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 ...
1
vote
0answers
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 ...
2
votes
2answers
43 views
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)...
0
votes
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,...
0
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0answers
27 views
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 $U:=\bigg\{\sum\limits_{1\leq i<j\leq 4} a_{ij}x_ix_j\mid a_{ij}\in\mathbb{C}\bigg\}$ ...
1
vote
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 ...
0
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0answers
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 ...
0
votes
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 =...
1
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0answers
51 views
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 $\sigma_2(n)+2\cdot \sigma(n) \le L(H(D_n))$?
...
0
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0answers
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 ...
1
vote
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 ...
0
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0answers
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?
1
vote
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 ...
2
votes
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 ,...
0
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1answer
43 views
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^...
5
votes
2answers
99 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 ...
3
votes
2answers
97 views
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
...
1
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2answers
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}},\...
0
votes
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?...
3
votes
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}...
3
votes
0answers
42 views
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 $f$ ...
1
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0answers
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 ...
0
votes
2answers
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 ...
0
votes
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 ...
2
votes
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 ...
1
vote
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 $$V_{k}=\exp({i\cdot\dfrac{2\pi k}{...
2
votes
1answer
46 views
Certain Isomorphic Representations of the dihedral group $D_{3}$
Using the following presentation of the dihedral group $D_{3}$
\begin{equation}
D_{3} =
\left\langle
r,s
\mid
r^{2} = s^{2} = (rs)^{3} = e
\right\rangle
\end{equation}
There is one (...
0
votes
2answers
27 views
Nonabelian dihedral groups and a question in number theory [duplicate]
I'll use a concrete definition of a dihedral group $D_{2n}$ which emphasizes its group structure:
$D_{2n}$ consists of distinct elements $r_0,...,r_{n-1},s_0,...,s_{n-1}$ so that for any $i \in \...
0
votes
1answer
61 views
Find representative for each conjugacy class of $D_{10}$
How do you find representative for each conjugacy class of $D_{10} = \langle r \rangle_5 \rtimes \langle s \rangle_2$?
I know $D_{10}$ has $4$ conjugacy classes which are:
$[Id]$, $[r]$, $[r^2]$, $...
-1
votes
1answer
47 views
Let $D_{2n}$ be the dihedral group of order $2n$. Let $H$ be the set of rotations of the regular $n$-gon. Is $H\lhd D_{2n}$? [closed]
Let $D_{2n}$ be the dihedral group of order $2n$, i.e., the group of symmetries of the regular $n$-gon.
Let $H$ be the set of rotations of the regular $n$-gon.
Is $H\lhd D_{2n}$?
1
vote
1answer
38 views
Compute a set $S$ given information about how it is acted upon transitively by $D_8$
Let $D_8=D_{2 \cdot 4}$ be the dihedral group on a regular $4$-gon. Suppose that $S$ is a subset of $S_4$, such that S contains the element $( 1 \ 2 \ 3)$. We also know that $D_8$ acts transitively ...
2
votes
1answer
84 views
How to show that $|D_{2n}| = 2n$ via the presentation?
Consider the dihedral group
$$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
...
2
votes
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 ...
0
votes
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 ...
3
votes
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 $...
1
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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 ...
0
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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 ...
4
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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 ...
1
vote
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)$...
0
votes
0answers
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)$
...
0
votes
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 ...
0
votes
0answers
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$...
1
vote
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 $...
1
vote
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 ...
1
vote
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 $...
3
votes
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. ...
0
votes
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 ...
