# 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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### Inner vs outer semidirect products of $S_3$ and $D_4$

I am trying to understand the difference between the inner and outer semidirect products of the symmetric group $S_3$ and the dihedral group $D_4$ of order $8$. The products are defined here. Inner ...
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### Finding subgroups of $D_{12}$

Let $G=D_{12}=\langle a,b \mid a^6=b^2=e, bab^{-1}=a^{-1} \rangle$. Find all subgroups of $G$. We can easily spot the cyclic normal subgroup $C=\langle a \rangle = \{e,a,a^2,\dots,a^5\}$. Now $C$ ...
1answer
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### Dihedral group is supersolvable

I need to show that Dihedral group $D_n$ is supersolvable. My Approach : I think the existence of a normal chain $\{e\} = G_0 \leqslant G_1 \leqslant ... \leqslant G_n = G$ satisfying following ...
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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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### 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 ...
1answer
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### 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 ...
0answers
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### 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 ...
0answers
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### Determine whether pairs of elements belong to the same left H-cosets

$G = D_6$ (dihedral of order 12, with generator $a$, $b$ where $a^6 = b^2 = e$ and $ba=a^{6−1}b$ $H=⟨a^1b^0⟩$ Given $(x,y)$ pairs, $(a,b^0), (b,a^5b), (b^0a^1, a^4)$, determine if $xH = yH$. The ...
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### Dihedral Groups; what exactly are the elements of the set?

I am reading Dummit and Foote. We have: [Definition 1:]For each $n \in \mathbb{Z}^+, n \ge 3$ let $D_{2n}$ be the set of symmetries of a regular $n-$gon, where a symmetry is any rigid motion of the ...
0answers
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### Why is the number of the conjugacy classes of $D_4 \rtimes Aut(D_4)$ $16$?

Currently, I am reading Representation Theory of Semi-Direct Products by Reyes. In section $6$, the author mentions that the number of the conjugacy classes of $D_4 \rtimes Aut(D_4)$ is $16$. My ...
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### Strategy for finding subgroups of the holomorph

I am trying to find subgroups (specifically dihedral subgroups of order $2p^2$) of $\text{Hol}(\mathbb{Z}/2p^2\mathbb{Z})$ (where $p$ is an odd prime), and have found it quite difficult to make ...
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### Computing the characters of $\prod^t_{i=1} S_{N_i} \wr D_{m_i}$

Let a group be $\prod^t_{i=1} S_{N_i} \wr D_{m_i}$ where $t \in \mathbb{Z} \text{ and } t \ge 1; N_i, m_i \in \mathbb{Z} \text{ and } N_i, m_i \ge 1$; and $S_{N_i}$ is a symmetric group over $N_i$ ...
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### Non-trivial rotation of dihedral group

Prove that the product of two distinct flips within the dihedral group is a non-trivial rotation. So this question is stating that if you have two different flips, that is the same thing as rotating ...
1answer
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### The order of $R^k \in D_n$, the $n$th dihedral group

I am using Thomas Judson's open-sourced Abstract Algebra textbook, available at http://abstract.ups.edu/aata, and I'm having trouble with this problem: Let $R$ represent a rotation element in $D_n$, ...
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### Graphs with automorphism $D_n \times G$ where $G$ is any finite permutation group

I would like to define a systematic process to construct graphs with automorphism $D_n \times G$ where $G$ is any finite permutation group. Here, $D_n$ is the dihedral group of order $2 n$. I follow ...
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### Determining the characters of an irrep of $D_4 \times D_8$

I am using Linear Representations of Finite Groups by Jean-Pierre Serre as my reference for this discussion. I would like to use the ideas from the book to determine the character of an irrep of the ...
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### An invertible, linear map that permutes a given set of points in $\mathbb{R}^2$

Let $V := \mathbb{R}^2$, $q \in \mathbb{N}$ and let a set $P := \{p_1,\dotsc,p_q\} \subseteq V$ be given. Define a set $\mathcal{S}$ where its members fullfill following properties: i.) S is an ...
1answer
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### Two reflections generating the Dihedral group $D_n$ [duplicate]

Let $l_1$ and $l_2$ be two lines intersecting at an angle of $\pi/n$ on $\mathbb{R}^2$. Let $r_1$ and $r_2$ denote the reflection by $l_1$ and $l_2$ respectively. I am to show that $r_1$ and $r_2$ ...
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### Geometrical Interpretation of quotient group of $D_{n}$ being isomorphic to $\{1,-1\}$

Recently I attempted the question Let $G$ be the dihedral group defined as the set of all formal symbols $x^{i}y^{j}$, $i=0,1,\ j=0,1,2,\ldots,n-1$ where $x^{2}=e,\ y^{n}=e,\ xy=y^{-1}x$. Prove (a)...
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### Automorphism groups of partially cycle graphs

I define partially cycle graphs as follows. If we add the same subgraph to $n-k$ vertices of an $n$-vertex cycle graph, where $1\le k < n$, we create a partially cycle graph. Here are a few ...
3answers
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### Number of orbits with Burnside's lemma

We color a equilateral triangle by coloring each edge with one of $k \geq 1$ colors. Find a formula for the number of orbits under the action of $D_6$, the dihedral group of $6$ elements, on the ...
1answer
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### What is the required group theory knowledge needed to understand Verhoeff's algorithm?

The Wikipedia page tells me I need to understand permutation groups and dihedral groups. Can someone clearly outline what exactly the perquisites of understanding this is and how much time I'll take ...
2answers
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### All groups of order 12; Dic12 and D6

I know this has been asked in various forms before, but so far I have failed to understand those answers properly. I've also read several papers discussing this, but I don't really get it. I have an ...
1answer
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### Classification of the irreducible group representations of the dihedral groups

Let $D_n$ be the dihedral group of order $2n$. Show that all irreducible representations have vector space dimension $1$ or $2$, and describe them up to isomorphism. Any hints how to even start?
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### Order of a certain finitely generated group

Suppose I am looking at the group $W=\langle s_\alpha, t_\beta \rangle$ where $s_\alpha$ and $t_\beta$ are reflections in $\mathbb{R}^2$ coming from two vectors $\alpha$ and $\beta$ making an angle of ...
1answer
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### Determine all the subgroups of the dihedral group $D_{15}$

Is there an algorithm for finding all of the subgroups of $D_{15}$? Also, is there a formula for finding the size of that subgroup? Not sure where to start with finding all the subgroups of $D_{15}$ ...
1answer
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### normaliser of dihedral group of order 4

Let $D_{2n}$ be the dihedral group of order $2n$. I would like to show that if $4$ divides $n$, then the normaliser $N_{D_{2n}}(D_{4})=D_{8}$. I know that $$|D_{2n}:N_{D_{2n}}(D_{4})|=|\mathcal{C}|,$$...
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### Multiplication table of $(S_1 \wr D_2) \times (S_1 \wr D_3)$

I am trying to compute the multiplication table of $(S_1 \wr D_2) \times (S_1 \wr D_3)$. My effort: I understand that $S_1$ is the trivial group consisting only the unit element i.e. $e$. $e$ will ...
0answers
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### Can there be a tower like $1 = A_0 \triangleleft \ldots \prod^t_i S_{N_i} \wr D_{m_i} \ldots \triangleleft = S_{\sum^t_i N_i m_i}$?

$\prod^t_i S_{N_i} \wr D_{m_i}$ is a subgroup of $S_{\sum^t_i N_i m_i}$. Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral ...
0answers
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### Interpreting $S_{N} \wr D_{m}$

I am trying to interpret $S_{N} \wr D_{m}$ in the light of the interpretation of $\mathbb{Z}^n_2 \wr \mathbb{Z}_2$ in this paper. So, according to the definition, \mathbb{Z}^n_2 \wr \mathbb{Z}_2 =...
1answer
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### When is the $k/n$ representation of $D_n$ irreducible, and why?

The $k/n$ representation of the Dihedral group of order $2n$ in $GL(2,\mathbb{C})$ is induced by mapping the rotation element of $D_n$ to the Rotation Matrix $R(\frac{2\pi k}{n})$, and the reflection ...