Questions tagged [dihedral-groups]
For questions on dihedral groups, the group of symmetries of a regular polygon, including both rotations and reflections
719
questions
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Binary sequences of $a$ zeros and $b$ ones which have $a+b$ circular shifts.
Let $P(a;b)$ be the set of all sequences generated by permutation of $a$ zeros and $b$ ones. $$|P(a;b)|=\binom{a+b}{a}$$
I want to prove that if $gcd(a,b)=1$, every sequence in $P(a;b)$ will have $a+b$...
- 766
1
vote
1
answer
46
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Proving a non-cyclic $G$ has a cyclic subgroup of index $2$ and $G\cong D_k.$
Suppose $G$ is a non-cyclic group generated by two elements, both of order $2.$ Prove that $G$ has a cyclic subgroup of index $[G:C]=2.$ If such a group is of finite order $2k,$ prove it is isomorphic ...
- 4,538
2
votes
0
answers
62
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Listing all subgroups of $D_{10}$
My work so far:
The cyclic subgroups are trivial to find. They are simply rotations generated from $\langle r^d\rangle$ where $d\mid10$. So $d=1,2,5,10$.
$\langle r \rangle= \{e,r, r^2, \dots, r^9 \}$
...
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1
vote
1
answer
27
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Showing that $f(r^k)^{-1}=f(r^k)$, where $f$ is a representation of the dihedral group $D_n$
Let $D_n$ be the dihedral group, which contains elements of the form $\{r^k\}_{k\in K}$ and $\{sr^k\}_{k\in K}$ where $K=\{1,\dots,n-1\}$ and $r$ and $s$ satisfy the equations
$$
r^n=s^2=1,\quad sr^ks=...
- 833
2
votes
0
answers
40
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Finding orbit in a group action of automorphism group of dihedral group on the dihedral group.
Finding orbit in a group action of automorphism group of dihedral group on the dihedral group.
Let $$D_{2n}=\langle a,b:a^n=b^2=1,bab^{-1}=a^{-1}\rangle $$ be the dihedral group of order $2n$ and $...
- 117
1
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2
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101
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No subgroup of $D_3$ has order 4.
Question: Prove that the dihedral group of order $6$ has no subgroup of order $4$.
I am trying to prove above question. But i have no idea how can i start! Just before this question (in exercise) i ...
- 516
3
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1
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117
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If $|G|=154$, and it not a direct product, then it must be isomorphic to $D_{154}$.
Suppose $|G|=154$ and $G$ is not decomposable in the sense that there are no proper subgroups $H$ and $K$ of $G$ such that $G \cong H \times K$. Show that $G \cong D_{154}$.
Here is my attempt: Let $P ...
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1
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1
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42
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Subgroup generated by two elements of group.
In the book Contemporary Abstract Algebra by Gallian. Chapter 3 (edition 9). Exercise 3 says
What can you say about a subgroup of $D_3$ that contains $R_{240}$ and a reflection $F$? What can you say ...
- 516
2
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2
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63
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Smallest $m$ such that $D_n$ is a subgroup of $S_m$
I was wondering what was, given any $n$ the smallest $m$ such that $D_n$ is a subgroup of $S_m$.
For cycle groups $C_n$ I believe there is a relation of the form $m = \min \sum_{k=1}^{K}p_k$ with :
$...
0
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0
answers
24
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What are the elements of the Dihedral Group $D_8$ when given by $D_8 = \langle a, b \mid a^2 = b^2 = (ab)^4=1\rangle?$ [duplicate]
I'm very confused since I've always seen the Dihedral Group $D_8$ given as
$$D_8 = \langle a, b \mid a^4 = b^2 =1, ab= a^3b\rangle.$$
I'm given the Definition
$$D_8 = \langle a, b \mid a^2 = b^2 = (ab)...
- 9
1
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0
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58
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Symmetries of the dihedral groups $D_n$ and $D_{nh}$.
According to my lecture book (Linear Representations of Finite Groups by Serre), the dihedral group $D_n$ consists of $n$ rotations and $n$ reflections in the plane that preserve a regular polygon ...
- 93
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0
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19
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Proving that $r^i \neq sr^j$ in $D_{2n}$ [duplicate]
I was trying to proof that the order of the dihedral group $D_{2n}$ is $2n$ given its presentation
$$
D_{2n} = \langle r,s ~\vert~ r^n = s^2 = 1 ,~ rs = sr^{-1} \rangle
$$
and I further assumed that ...
1
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0
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42
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Importance of abelian subgroups of dihedral group in computing conjugacy classes?
Let $D_{8}$ be the dihedral group of order 8. We would like to find its conjugacy classes. Authors Dummit and Foote write:
In $D_{8}$ we may also use the fact that the three subgroups of index 2 are ...
- 759
3
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1
answer
56
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Unique $p$-Sylow subgroup of Dihedral group $D_n$.
If $p>2$ is a prime number that divides $n \in \mathbb N$, then $D_n = \langle r,s \rangle$ (with $r^n=s^2=id, sr=r^{-1}s$) has a unique $p$-Sylow subgroup $H \leq D_n$.
I'm having trouble with ...
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3
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0
answers
29
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Embedding $D_{2n}$ in $U_2$
I have a problem doing Exercise 18.8 of groups and symmetry by M. A. Armstrong.
18.8. Look at our description of the Möbius band as a subset of $\mathbb{C} \times \mathbb{C}$ and find matrices in $...
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125
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When does $|G|=pq$ admit unique non abelian representation? [duplicate]
Classifying $G$ with $|G|=pq$ has been asked several times in this site, but I am having trouble with number theory. Let $q<p$, $Q=\langle u\rangle\in \text{Syl}_q(G)$, $P=\langle v\rangle\in \text{...
- 1,005
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37
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What do the dotted lines represent in the hexagonal lattice system?
The above is one of the four two-dimensional lattice systems. See this article for some info.
The parallelogram is the primitive unit cell, because it can be translated in order to recreate the entire ...
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1
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108
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Representation of symmetry group $D_4$ acting on a square
Consider the square with vertices $(\pm 1, \pm 1)$. The symmetry group $D_4$ of this square acts by permuting the vertices
Show that each permutation comes from a linear transformation. Compute the ...
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28
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Let $n=dm$ and consider the group $D_n$ , with $K=⟨ρ^m⟩$ .Show that the following $2m$ cosets of $K$ are all distinct:
$D_n = \langle \mu, \rho \mid \mu^2 = \rho^n = \epsilon, \mu\rho = \rho^{-1}\mu \rangle$
Let $n=dm$ and consider the group $D_n$, with $K = \langle \rho^m \rangle$. Show that the following $2m$ cosets ...
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How do i show that $\psi : D_n \rightarrow D_m$ defined by $\psi(\mu^i \rho^j) = \delta^i \gamma^j$ is a homormophism from $D_n$ to $D_m$ where n =qm
Let $n = dm$
Consider these groups:
$D_n = \langle \mu, \rho \mid \mu^2 = \rho^n = \epsilon, \mu\rho = \rho^{-1}\mu \rangle$
$D_m = \langle \delta, \gamma \mid \delta^2 = \gamma^m = \epsilon, \delta\...
- 751
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1
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Definition and Properties of Dihedral Group
Definition: Here is equivalent definition of dihedral group $$\begin{align}D_{2n}&=\{\sigma \in S_n\mid \sigma(a+1)\equiv\sigma(a)+1\pmod n, \forall 1\leq a\leq n\} \bigcup \{\sigma \in S_n\mid \...
- 3,619
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2
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73
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Help understanding how to show that two groups are isomorphic
I'm trying to understand what is required to show that two groups are isomorphic. My understanding is that two groups $G_1$ and $G_2$ are isomorphic if:
$ |G_1| = |G_2| $ (equal cardinality)
$G_1$ ...
0
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1
answer
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Definition of Dihedral Group in Dummit’s Abstract Algebra
For each $n\in \Bbb{Z}^+$, $n\geq 3$ let $D_{2n}$ be the set of symmetries of a regular $n$-gon, where a symmetry is any rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-...
- 3,619
4
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0
answers
107
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Geometrical interpretation of the faithful actions of $D_6$ and $D_{10}$ on sets of size $5$ and $7$, respectively. [duplicate]
The dihedral group of order $2n$, $D_n$, acts faithfully on the set of the $n$ vertices of the regular $n$-gon. For $n=6$ and $n=10$, $D_n$ acts faithfully also on sets of size smaller than $n$, ...
- 2,365
2
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3
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70
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The dihedral group $D_n$ cannot be written as a product of two nontrivial groups
Let $n$ be an odd prime. I want to prove that the dihedral group $D_n$ cannot be written as a product of two nontrivial groups. But it seems problematic, as I completely neglect the properties of ...
- 367
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0
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37
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Understanding cycle notation for a dihedral group and one of its subgroups
I'm looking at an example that instructs me to consider the group:
$$
G = \langle(1, 2, 3, 4, 5, 6, 7, 8), (1, 8)(2, 7)(3, 6)(4, 5)\rangle \cong Dih(16)
$$
And the subgroup:
$$
H = \langle(1, 3, 5, 7)(...
- 67
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0
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40
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Isomorphisms between subgroups of the Symmetry group of the cube
I need to list all the subgroups of the symmetry group of the cube that are isomorphic to either $\mathbb{Z}_n$ or $\mathbb{D}_n$ (the dihedral group) for some $n \in \mathbb{N}$.
I know that the ...
- 353
1
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0
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Prove $D_6 \cdot \text{stab}_{S6}(2) \leq S_6.$
I wish to show $D_6 \cdot \text{stab}_{S6}(2) \leq S_6$, where $D_6$ is the dihedral group of order $12$ and $S_6$ is the symmetric group of order $6$.
I have been able to provide the start of a ...
- 145
1
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2
answers
71
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Is the following presentation for $D_8$ a valid presentation?
My professor used the following presentation for $D_8:$
$$D_8 = \{s,t | s^2 = t^2 = (st)^4 = 1\}$$
But I am not sure if this presentation is correct, I looked at subWiki here https://groupprops....
- 3,429
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1
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In group theory, are the operations left to right or right to left?
I got points taken off on homework because for the dihedral group $D_3$, I needed to find the left coset of a subgroup $H$. So for "flip $\cdot$ rotation" I did the flip first and then the ...
2
votes
1
answer
75
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Describing holomorph of dihedral group
Are there any articles or books, where ${\rm Hol}(D_{2n}$) ($D_{2n}$ is dihedral group of order $2n$), is described?Semidirect product seems like very hard thing to compute, but I'm interested whether ...
- 367
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Are the mean and variance of a dihedral angle periodic?
In biomolecular science, a dihedral angle is periodic with a period of $2\pi$. It ranges from -180 to 180 degrees. Now, if for 5 dihedral angles, I want to calculate their mean and variance. Will the ...
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1
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Looking for a subgroup $H$ of $D_{2019}$ with the following properties:
I m looking for a subgroup $H$ of the dihedral group $(D_{2019},\circ)$ so that:
With $(D_{2019},\circ)$ the group containing as elements
$b$: a reflexion on a symmetry axis, and $a$: a rotation of $\...
- 113
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2
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59
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Let $a,b\in D_4$ and such that $ba=a^{3}b$. Prove that if $0\leq i<4$ and $0\leq j<2$, then $a^ib^ja^{i_1}b^{j_1}=a^{i+3i_1}b^{j+j_1}$
Let $a,b\in D_4$ and such that $ba=a^{3}b$. Prove that if $0\leq i<4$ and $0\leq j<2$, then $a^ib^ja^{i_1}b^{j_1}=a^{i+3i_1}b^{j+j_1}$,(where $i,j,i_1,j_1\in \mathbb{Z}$) also it is given $o(a)=...
- 2,556
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Step in a proof that $G:=\langle x,y \mid x^n,y^2, (xy)^2\rangle$ is isomorphic to the dihedral group $D_n$
I’m reading a proof of the fact that the group given by the presentation $G:=\langle x,y \mid x^n,y^2, (xy)^2\rangle$ is isomorphic to the dihedral group $D_n$. It begins like this:
Let $G=\langle \...
- 411
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2
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Prove that all dihedral groups of degree $4$ are isomorphic
I was studying about group theory when I came across a very well-known statement that:
Any two dihedral groups of degree $4$ are isomorphic.
I have two questions from here:
What’s actually is a ...
- 2,556
4
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1
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Show that the $D_{3h}$ prism is given by the direct product $D_{3h}=D_3 \otimes C_s$
I have some questions regarding the notation and solution to the following problem involving the direct product:
The diagram below shows the symmetry operations of an equilateral right triangular ...
- 59
1
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0
answers
44
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Fitting length of Dihedral groups
Let $G$ be a dihedral group of order $2n$ where $n\geq 1$, denoted by $D_n$. We know that $G$ is nilpotent if and only if $n=2^i$ for all $i\geq 1$, a proof of this you can check in the below link 1. ...
- 191
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1
answer
40
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Show that $D_{2^n}/Z(D_{2^n})\cong D_{2^{n-1}}$.
The Problem: Show that $D_{2^n}/Z(D_{2^n})\cong D_{2^{n-1}}$.
I know that $Z(D_{2^n})=\{1, r^{2^{n-1}}\}$, so I figured that establishing an explicit isomorphism $\phi: D_{2^n}/Z(D_{2^n})\to D_{2^{n-1}...
- 1,179
3
votes
1
answer
95
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Embedding of dihedral group in $\mathbb{Z}_{2^n}$
Let $D_{2^n}$ be a dihedral group of order $2^n$. I want to find embedding of $D_{2^n}$ in $GL(2,\mathbb{Z}_{2^n})$ if such embedding exists. I know, that $D_{2^n}$ is isomorphic to the next subgroup ...
- 367
2
votes
2
answers
126
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Question on the definition of the Dihedral groups
I have seen the definition of the dihedral group of order $2n$ in several places as
$$
D_n = \langle x,y \mid x^n = y^2 = e, yxy^{-1} = x^{-1}\rangle.
$$
My question is why there is an "inverse&...
- 3,003
0
votes
0
answers
47
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Doubts in the action of $D_4$ on the set of $2$-colorings of a square.
Want to find the action of the group $D_4$ on the set of $2$-colorings of square, as per the source. It states, in section 4.1, on page #12.
Kindly vet the action of symmetry elements on the $2^4=16$ ...
- 4,480
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votes
1
answer
50
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Find different symmetries of $D_4$ by composition of component symmetries.
Need find the different symmetries of $D_4$ by composition.
Kindly tell if is the below process of deriving symmetries' permutations correct.
Let, $[1234]$ denote the initial configuration of square ...
- 4,480
1
vote
1
answer
27
views
Show that for even integers $ n $ there exists an element $ g \in D_{n} $ : $ \operatorname{ord}(g)=2 $ and $ \operatorname{sgn}(\varphi(g))=1 $
Let $ n>3 $, let $ \Delta_{n} $ be a regular $n$-corner, and let $ D_{n} $ be the dihedral group which is the symmetry group of $ \Delta_{n} $. If we number the vertices of $ \Delta_{n} $, we ...
- 560
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votes
0
answers
37
views
Find the cycle index of $C_{10}, D_{10}$ acting on the vertices of the regular $10$-gon.
Find the cycle index of :
(a) $C_{10},$ and
(b) $D_{10}$
acting on the vertices of the regular $10$-gon.
(a) There are ten rotations, and
$r, r^{-1}$ both equal to $x_{10}^1,$ as one cycle of order $...
- 4,480
4
votes
3
answers
137
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Sylow-$2$ Subgroups of a simple group of order 168
I'm stuck somewhere in the following claim, I would appreciate if you could help:
Claim: Let $G$ be a simple group of order $168(=2^3\cdot 3\cdot 7).$ Then all Sylow $2$-subgroups of $G$ are dihedral.
...
- 407
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1
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303
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Consider a reflection as an element of the permutation group of the vertices of a regular (2n + 1)-gon. Is the permutation even or odd?
Let $n \ge 1$ be an integer. Let $F \in D_{(2n+1)}$ be a reflection. Consider $F$ as an element of the permutation group of the vertices of a regular $(2n + 1)$-gon. Is this permutation even or odd?
...
0
votes
1
answer
61
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Conjugacy classes of subgroups of $D_{4}$
This problem is from Michael Artin Algebra first edition.
6.3.3) List all subgroups of the dihedral group $D_{4}$ and divide them into conjugacy classes.
I am a bit unsure what I am being asked here ...
- 686
1
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1
answer
37
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Intersection of two cyclic groups in $D_{n}$
For the dihedral group $D_{n}=<\rho,\tau | \rho^{n}=\tau^{2}=id, \rho\tau=\tau\rho^{-1}>$
How can we prove that the intersection of $<\rho>$ and $<\tau>$ is the trivial group $\{e\}$,...
- 433
1
vote
0
answers
141
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Generators and relations of the dihedral group
From Aluffi's Algebra: Chapter $0$
Describe generators and relations for all dihedral groups $D_{2n}$. (Hint: Let $x$ be the reflection about a line through the center of a regular n-gon and a vertex,...
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