Questions tagged [dihedral-groups]
For questions on dihedral groups, the group of symmetries of a regular polygon, including both rotations and reflections
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 ...
2
votes
2answers
32 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
25 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 $$V_{k}=\exp({i\cdot\dfrac{2\pi k}{...
2
votes
1answer
42 views
Certain Isomorphic Representations of the dihedral group $D_{3}$
Using the following presentation of the dihedral group $D_{3}$
\begin{equation}
D_{3} =
\left\langle
r,s
\mid
r^{2} = s^{2} = (rs)^{3} = e
\right\rangle
\end{equation}
There is one (...
0
votes
2answers
24 views
Nonabelian dihedral groups and a question in number theory [duplicate]
I'll use a concrete definition of a dihedral group $D_{2n}$ which emphasizes its group structure:
$D_{2n}$ consists of distinct elements $r_0,...,r_{n-1},s_0,...,s_{n-1}$ so that for any $i \in \...
0
votes
1answer
35 views
Find representative for each conjugacy class of $D_{10}$
How do you find representative for each conjugacy class of $D_{10} = \langle r \rangle_5 \rtimes \langle s \rangle_2$?
I know $D_{10}$ has $4$ conjugacy classes which are:
$[Id]$, $[r]$, $[r^2]$, $...
0
votes
1answer
37 views
Let $D_{2n}$ be the dihedral group of order $2n$. Let $H$ be the set of rotations of the regular $n$-gon. Is $H\lhd D_{2n}$?
Let $D_{2n}$ be the dihedral group of order $2n$, i.e., the group of symmetries of the regular $n$-gon.
Let $H$ be the set of rotations of the regular $n$-gon.
Is $H\lhd D_{2n}$?
1
vote
1answer
34 views
Compute a set $S$ given information about how it is acted upon transitively by $D_8$
Let $D_8=D_{2 \cdot 4}$ be the dihedral group on a regular $4$-gon. Suppose that $S$ is a subset of $S_4$, such that S contains the element $( 1 \ 2 \ 3)$. We also know that $D_8$ acts transitively ...
2
votes
1answer
77 views
How to show that $|D_{2n}| = 2n$ via the presentation?
Consider the dihedral group
$$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
226 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 ...
4
votes
1answer
39 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
vote
1answer
30 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
votes
1answer
31 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
votes
1answer
63 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
19 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
12 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
21 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
50 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
44 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
81 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
62 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
38 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 ...
1
vote
2answers
34 views
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 ...
1
vote
2answers
81 views
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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0answers
32 views
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 ...
1
vote
2answers
44 views
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 ...
-1
votes
1answer
29 views
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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vote
2answers
27 views
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}...
1
vote
2answers
22 views
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 ...
1
vote
1answer
16 views
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 \...
2
votes
1answer
33 views
$G_{x_1x_2+x_3x_4}\cong D_8$
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\...
0
votes
2answers
57 views
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 ...
1
vote
2answers
59 views
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 ...
0
votes
0answers
76 views
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.
5
votes
1answer
85 views
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 ...
2
votes
1answer
57 views
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$ ...
1
vote
1answer
21 views
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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votes
1answer
46 views
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 ...
1
vote
1answer
16 views
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,...
0
votes
1answer
32 views
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}\}$
...
1
vote
0answers
21 views
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 ...
0
votes
1answer
20 views
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 & ...
1
vote
1answer
15 views
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\...
4
votes
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 $K$ ...
0
votes
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....
0
votes
0answers
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 ...
0
votes
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/...
1
vote
1answer
35 views
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 ...
