Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [dihedral-groups]

For questions on dihedral groups, the group of symmetries of a regular polygon, including both rotations and reflections

2
votes
1answer
30 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
16 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 ...
-1
votes
1answer
41 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
11 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
17 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
16 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
13 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
13 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
116 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
29 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
25 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
29 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
22 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 ...
1
vote
2answers
42 views

conceptual question with example: proving groups are isomorphic

The question is to prove $D_8$ and the subgroup of $S_4$ generated by $(1 2)$ and $(1 3)(2 4)$ are isomorphic. I was able to show that the relations for $D_{8}$ follow when we set $b = (1 2)(1 3)(2 4)...
2
votes
1answer
51 views

Is there a way to describe all finite groups $G$, such that $Aut(G) = D_4$?

Is there a way to describe all finite groups $G$, such that $Aut(G) = D_4$? Two groups, that definitely satisfy that condition are $D_4$ itself and $\mathbb{C}_2 \times \mathbb{C}_4$. I have read ...
2
votes
0answers
39 views

Minimal amount of information required to define a Dihedral group?

Given the dihedral group $D_n$, where $r$ is a single rotation, and $s$ is a reflection, I must show that $s \circ r \circ s = r^{-1}$. $D_n$ is a group, with properties: 1) $1 = r^0 = s^0$ 2) for ...
2
votes
1answer
28 views

Orbit of conjugation on subgroups of $D_8$

Let $X$ be the set of all subgroups of $D_8$ with order $2$. For fixed $g\in D_8$, and for all $x\in X$, conjugation by $g$ is defined by $$x\mapsto gxg^{-1}$$ What is the orbit of this group action? ...
-1
votes
1answer
38 views

Group G with elements of specific order [closed]

What is an example of a group G with elements a and b such that the order or a and b is 2, but the order of ab is 3? I'm thinking some sort of Dihedral group perhaps?
1
vote
0answers
42 views

Is it possible to draw a non-square of symmetry group $D_8$?

Just like the question says. I was trying to find four different shapes but the only one that I could think of is a rhombus but it would only have 2 reflections, not 4. Is this even possible?
0
votes
0answers
32 views

Prove that $D_{2n} = \langle s, rs \rangle$

Show that the subgroup of $D_{2n}$ generated by the set $\{s,rs\}$ is $D_{2n}$ itself. Here is my attempt: $\langle s, rs \rangle = D_{2n}$ if and only if $\langle r, s \rangle = \langle s, rs \...
3
votes
2answers
33 views

Find all characteristic subgroups of the Dihedral group $D_{12}$.

In my notation $$D_{12}=\langle \rho,\tau : \rho^6=\tau^2=1,\ \rho \tau \rho=\tau\rangle$$ So firstly, I know that all characteristic subgroups are normal. Thus, the possible candidates of $D_{2n}$ ...
3
votes
1answer
51 views

Non cyclic group of order $8$ having exactly one element of order $2$

Let $G$ be a non-cyclic group of order $8$ having exactly one element of order $2$. Prove that $G$ is generated by elements $a$ and $b$ subject to the relations $a^4=1$ and $a^2=b^2$. I can start ...
0
votes
1answer
39 views

Composition series for dihedral grup of order $2pq^2$

Let $p,q$ be different primes and $D_{2pq^2}=\langle ab\mid a^{pq^2}=b^2=1, ba=a^{-1}b\rangle$ the dihedral group of order $2pq^2$. Find a composition series for $D_{2pq^2}$. I don't know how to ...
1
vote
2answers
65 views

How many distinct composition series does the group $D_{12}$ have?

How many distinct composition series does the group $D_{12}$ have? I know that $D_{12} \trianglerighteq \mathbb{Z}_6 \trianglerighteq \mathbb{Z}_3 \trianglerighteq \{e\}$ is a composition series ( ...
3
votes
1answer
55 views

Groups of order $2p$

There are a few question on classification of groups of order $2p$ on MSE but I'd like to receive a feedback on this proof (and have a question about it at the end). Let $G$ be a group of order $2p$. ...
1
vote
2answers
38 views

Color Vertices of a Regular Pentagon with Two Colors

I'm trying to figure out how many distinct ways there are to color the five vertices of a pentagon two different colors. I know this requires the use of Burnside's theorem, but am struggling a bit ...
3
votes
1answer
71 views

Prove that $| \operatorname{Aut}(D_n)|\le n\phi(n)$

Prove that for $n\gt 2$, $| \operatorname{Aut}(D_n)|\le n\,\phi(n)$ where $D_n$ is the dihedral group with 2n elements and $\phi$ is Euler phi function. Let $\rho$ be a rotation such that $o(\rho)=n$,...
2
votes
0answers
75 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 ...
2
votes
1answer
56 views

Counting elements of order 6 in $D_{12} \times Z_2$

Actually I know how to count number of elements of particular order from direct product .But In this my counting is not matches with answer so I wanted to know where is my mistake lies . I wanted to ...
1
vote
2answers
77 views

Is it only the generator of the group that commutes with all the other elements?

If a group is generated by an element does that mean the generator commutes with all the other elements or does it mean that because the group is cyclic(as it has a generator) that all elements ...
0
votes
2answers
39 views

Show that the dihedral group $D_{16}$ is the internal direct product of its Sylow subgroups.

Show that the dihedral group $D_{16}$ is the internal direct product of its Sylow subgroups. (We use the notation $D_{16}$ for the dihedral group of order 32) Here's what I think. Since $D_{16}$ is ...
3
votes
2answers
65 views

What is Gal$_\mathbb{Q}(x^4 + 5x^3 + 10x + 5)$?

Find Gal$_\mathbb{Q}(x^4 + 5x^3 + 10x + 5)$ and Gal$_\mathbb{Q}(x^4 - 2)$ I was trying the second one which I think is the easiest case. However I am not able to prove it. Here is what I know (1) ...
0
votes
1answer
28 views

How does the element $ ba^{n} $ become $a^{3n}b $ from the relation $ ab=ba^{3}$ of the group $ D_{4}$?

moving $a^{n}$ past b it makes sense for me for this to become $a^{n/3}b $ rather than $a^{3n}b$ from the given relation. Am I missing something obvious here ?
4
votes
0answers
53 views

Principled way to find a shape with symmetries given by a group

Recently I've learned how groups correspond to symmetries of objects so I've been trying to find shapes corresponding to groups that I know (all with finite groups). For example, I know $\mathbb Z / ...
1
vote
0answers
98 views

Prove or disprove: If $H$ is normal in $G$ and $H$ and $G/H$ are abelian, then $G$ is abelian. [duplicate]

My counterexample I wanted to try is $D_{8}$ and its center $Z(D_{8})$. However, is $Z(D_{8})$ abelian? I am guessing it is because it is cyclic? I am not very knowledgeable with dihedral groups.
2
votes
0answers
45 views

Find a topological space whose fundamental group is $D_4$ [duplicate]

$D_4$ here indicates the dihedral group of order 8. Does there exist any trivial example of topological spaces such that it has $D_4$ as it's fundamental group.
1
vote
2answers
46 views

Show that a dihedral group of order $4$ is isomorphic to $V$, the $4$ group. [closed]

Show that a dihedral group of order $4$ is isomorphic to $V$, the $4$ group. Also, please show that a dihedral group of order $6$ is isomorphic to $S_3$. Thank you!
2
votes
1answer
38 views

Is there a dihedral graph in which the vertices have degree 4?

Looking at the graphs of diheral groups I noticed most have vertices of degree $2$ or $3$. The reason might seem obvious but in mathematics I have learned not to assume anything! Is there such a ...
2
votes
1answer
90 views

Is there a non-trivial homomorphism from $D_4$ to $D_3$?

The question is actually a bit broader than this, but it is a good starting point. Given two Dihedral groups $D_4$ and $ D_3$ we wish to construct a nontrivial homomorphism $f:{ D_4\to\ D_3}$. Is ...
0
votes
0answers
24 views

Number of congruences for given polyhedron

First: I don't know if I'm using the terminology in the right way, since I translated this. In my textbook, a congruence is a transformation that keeps all distances equal, and it is proved that ...
2
votes
1answer
45 views

Show that no group has $D_n$ as its derived subgroup.

(Yesterday I found an answer to this exact question by mistake and today I am not being able to find it again...) I am asked to show that no group has $D_n$ as its derived subgroup. I am also given a ...
0
votes
0answers
29 views

Normalizer of a Sylow 2-subgroup of dihedral group $D_{2n}$

I'm trying to prove the following result: if $P\in Syl_2(D_{2n})$ then $N_{D_{2n}}(P)=P$. That is, the normalizer of a Sylow 2-subgroup in $D_{2n}$ is itself.
1
vote
1answer
57 views

What is the center of $D_{2n}/Z(D_{2n})$

What is the center of $D_{2n}/Z(D_{2n})$. I see that when $n=2^k$ then I have a $p$-group so the center is not trivial. But when $n$ is not power of $2$ how can i know what is the center of this group?...
0
votes
0answers
19 views

The centralizers of all elements of group $G$ when $\frac{G}{Z(G)}\cong D_{2m}$

Let $G$ be a finite group so that $\frac{G}{Z(G)}\cong D_{2m}$, for $m\geq 2$. What are all centralizers of elements of group $G$? Assume that $\frac{G}{Z(G)}\cong \{ xZ(G),yZ(G):x^2 ,y^m ,xyx^{-...
2
votes
2answers
61 views

Presentation of $D_4$

When one writes $D_4=\langle r,s\mid r^4=s^2=1,rsrs=1\rangle$ they are describing a quotient group. Let $S=\{s,r\}$ and $R=\{r^4,s^2,rsrs\}$. $$F_S=\langle r,s\rangle,\quad R^{F_S}=\{grg^{-1}\mid r\in ...
1
vote
0answers
33 views

Showing a map is well defined (Mobius map to $D_{2n}$ in group theory)

I am asked to show that the subgroup $G$ of the Mobius group $M$, generated by $f(z)=e^{2\pi I/n}z$ and $g(z)=\frac{1}{z}$ is isomorphic to $D_{2n}$. I considered the mapping $$h: G \to D_{2n} \text{...
0
votes
0answers
53 views

Terminology for dihedral groups

What notation is most common for the dihedral group of order $2n$? I'm talking about the group of symmetries of a regular $n$-gon. I know that some books call this group $D_n$, and some books call it $...
1
vote
0answers
40 views

Show that $D_3\times_\rho\mathbb{Z}_2$ is not isomorphic to $A_4$

This is the third part of a problem. I will list here the first two parts as a reference and then my attempt to solve the third one. I want to verify if the first and second are right and some help ...
1
vote
1answer
306 views

Number of conjugacy classes of a Dihedral group?

How do you find the number of conjugacy classes of a Dihedral group? Say for D11 for example. I know by Lagrange each conjugacy class has order 1, 2, or 11. For smaller n, it can sometimes just be ...
1
vote
0answers
140 views

Number of group homomorphisms from infinite cyclic group to dihedral?

Want to determine the number of group homomorphisms $f: \mathbb{Z} \to D_7$. My guess is that there is only $1$ because $0$ is the only element with finite order in $\mathbb{Z}$. Note a cyclic group ...