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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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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 ...
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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 $D_{2n}=\langle\sigma ,\tau|\sigma^n=\tau^2,\sigma^{\tau}=\sigma^{-1}\rangle$...
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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 $...
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1answer
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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 ...
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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 $...
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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. ...
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1answer
34 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 ...
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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 ...
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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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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 ...
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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 ...
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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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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}...
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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 ...
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1answer
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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 $a^ib^j, \ 0 \...
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$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\...
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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 ...
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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 ...
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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.
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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 ...
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1answer
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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 $x$ ...
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1answer
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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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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 ...
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1answer
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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,...
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1answer
24 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 $D_n=\{\langle a,b\rangle :a^n=b^2=1,bab=a^{-1}\}$ ...
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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 ...
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1answer
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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 & ...
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1answer
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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\...
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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 $K$ ...
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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....
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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 ...
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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/...
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1answer
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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 ...
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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)...
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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 ...
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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 ...
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1answer
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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? ...
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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?
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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 \...
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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 $D_{2n}$ ...
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1answer
61 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 ...
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1answer
55 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 ...
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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 ( ...
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60 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 $2p$. ...
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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 ...
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1answer
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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 $o(\rho)=n$,...
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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 ...
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1answer
65 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 ...
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2answers
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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 ...
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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 ...