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

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18 views

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 ...
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1answer
42 views

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 $D_{15}$ ...
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1answer
14 views

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 ...
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19 views

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 ...
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10 views

Maximal set of disjoint prime cycle permutations on $n$ elements to generate $\prod^t_i S_{N_i} \wr D_{m_i} $

How can I determine the maximal set of disjoint prime cycle permutations on n elements to generate $\prod^t_i S_{N_i} \wr D_{m_i} $? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the ...
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1answer
19 views

Are subgroups of automorphism groups of trees direct product of symmetric, cyclic and dihedral groups?

My question is triggered by my confusion with the notation $\Psi$ in Constructive Approach to Automorphism Groups of Planar Graphs by Klavík et al. The notation $\Psi$ was first used expressing ...
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8 views

Relation between $\prod^t_i S_{N_i} \wr D_{m_i} $ and $A_{\sum^t_i N_i m_i}$?

$\prod^t_i S_{N_i} \wr D_{m_i}$ is a subgroup of the symmetric group $S_{\sum^t_i N_i m_i}$. Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ ...
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8 views

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 ...
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14 views

Is $\prod^t_i S_{N_i} \wr D_{m_i}$ a maximal subgroup of $S_{\sum^t_i N_i m_i}$?

How can I determine whether $\prod^t_i S_{N_i} \wr D_{m_i}$ is a maximal 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 $N_i$ ...
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0answers
6 views

Presentation of $\prod^t_i S_{N_i} \wr D_{m_i}$

I would like to determine the presentation of $\prod^t_i S_{N_i} \wr D_{m_i}$. My effort: The presentation of the symmetric group $S_{N}$ is as follows. $\langle s_1, \ldots, s_{N-1} | (s_i ...
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0answers
16 views

Is $\prod^t_i S_{N_i} \wr D_{m_i}$ a normal subgroup of $S_{\sum^t_i N_i m_i}$?

How can I determine whether $\prod^t_i S_{N_i} \wr D_{m_i}$ a normal 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 $N_i$ ...
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1answer
46 views

What are the elements of $D_\infty$?

Let $f$ be a function of the real line to itself which preserves distance and sends integers to integers. (i) Assuming that $f$ has no fixed points, show that $f$ is a translation through an integral ...
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12 views

How to determine the order of $\sum^t_i S_{N_i} \wr D_{m_i}$?

How can I determine the order of $\sum^t_i S_{N_i} \wr D_{m_i}$ ? 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 group of ...
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0answers
10 views

How to prove $\sum^t_i S_{N_i} \wr D_{m_i}$ non-Abelian?

How do I prove that $\sum^t_i S_{N_i} \wr D_{m_i}$ is a non-Abelian group? 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 ...
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2answers
22 views

Do the elements $rs$ and $r^2s$ generate the dihedral group $D_n$?

The dihedral group $D_n$ is generated by the elements $r$ and $s$. Is it possible for the elements $rs$ and $r^2s$ generate the group as well?
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1answer
24 views

Conjugacy classes, irreducible representations, character table of $D_{10}$ (order 20)

$D_{10}=\langle r,s \rangle$ is the dihedral group of order 20 . I have been struggling a bit with this question, particularly c, regarding values in the character table (a). Find the conjugacy ...
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1answer
12 views

Given the lattice, find all pairs of elements that generate $D_{8}$

Given the subgroup lattice of the dihedral group $D_{8}$, find find all pairs of elements that generate $D_{8}$. There should be 8 pairs. The given lattice is the image below: I came to following ...
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1answer
37 views

The center of the dihedral group [closed]

How to prove that the center of the dihedral group $D_{2n}$ is $\{1,r^{n}\}$ and the center of $D_{2n-1}$ is $\{1\}$? I don't know how to prove it in this general case.
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1answer
270 views

Is a group defined by its generator set and relations?

I'm learning about generators from Dummit and Foote. They call this a presentation of the dihedral group: $$D_{2n} = \left< r,s\,|\, r^n=s^2=1,\, rs=sr^{-1}\right>$$ Does this type of ...
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1answer
82 views

The smallest symmetric group $S_m$ into which a given dihedral group $D_{2n}$ embeds

Several questions, both here and on MathOverflow, address the issue of determining for a given group $G$ the smallest integer $\mu(G)$ for which there is an embedding (injective homomorphism) $G ...
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2answers
58 views

Prove G has a nontrivial center

Let $G$ be a group which has a normal subgroup isomorphic to $D_8$. Prove that $G$ has a non trivial center. So, given $g\in G$, $h\in D_8$ $ghg^{-1}\in D_8$. So I tried to prove that there is an ...
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2answers
36 views

How are the elements of a dihedral group usually defined?

While searching online, I've come across two ways to define the elements of the dihedral group. Both ways are internally consistent and are fine as far as I can tell, but they are mutually exclusive, ...
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1answer
27 views

Dihedral groups are solvable [duplicate]

I'm trying to prove the dihedral groups are solvable for any Dn. I use the normal subgroup of all rotations, since the quotient of Dn/{rotations} is isomorphic to Z2 so it's abelian as well, so we get ...
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0answers
17 views

Exponent of the direct sum of finite groups, specifically, $\sum^t_i S_{N_i} \wr D_{m_i}$

I have one general and one specific questions. What is the expression for the exponent of the direct sum of finite groups? What is the exponent of $\sum^t_i S_{N_i} \wr D_{m_i}$? Here, $i, N_i, ...
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12 views

Relation between direct product and direct sum of $S_{N_1} \wr D_{m_1}, \ldots, S_{N_t} \wr D_{m_t}$

I am trying to understand the relation between the direct sum and direct product of all the groups from the set $$\{S_{N_1} \wr D_{m_1}, \ldots, S_{N_i} \wr D_{m_i}, \ldots, S_{N_t} \wr D_{m_t}\}$$ ...
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0answers
64 views

What can we say about $D_{2n}$ and $D_n$ if $n$ is even?

We know that if $n$ is an odd natural number then the dihedral group $D_{2n}$ is isomorphic to the group $D_n\times \mathbb{Z}_2$ where $D_m$ is the dihedral group of order $2m$, which has ...
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2answers
38 views

Finding $N(D_{4})/D_{4}$ for $D_{4}$ in $D_{16}$

I want to find $N(D_{4})/D_{4}$ where $N(D_{4})$ is the normalizer of $D_{4}$ in $D_{16}$. I'm not too clear on what the normalizer of $D_{4}$ in $D_{16}$ Is there a nice way to find ...
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1answer
32 views

How to write dihedral group in cycle notation?

Since each symmetry can be thought of as a permutation of the vertices, the elements of $D_n$ can be thought of as elements of $S_n$. So I'm wondering if there's a systematic way that we can always ...
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0answers
12 views

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 ...
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3answers
46 views

Let $n \ge 3$. Prove that there are $n!$ different one-to-one homomorphisms from $D_n$ to $S_n$

Let $n \ge 3$. Prove that there are $n!$ different one-to-one homomorphisms from $D_n$ to $S_n$. I know there are $n!$ elements in $S_n$, but this fact didn't get me anywhere. I tried many ...
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0answers
28 views

What is the real life interpretation of the number of orbits in a bracelet problem?

I was asked to find the number of necklaces that can be made using 4 red beads and 6 blue beads. I understand the steps needed to be taken in order to solve this type of problem but I don't quite ...
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1answer
43 views

How do I find all of the orbits and stabilisers of X?

Consider $D_{10}$ The group of symmetries of the regular pentagon. Let $\sigma= (12345)$ and $\tau =(13)(45)$ being rotation by $72^{\circ}$ and reflection (with 2 being the fixed point) respectively. ...
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2answers
72 views

Group Theory: How to find all possible images of $ f $?

Let $H$ be a group and suppose that $ f: D_{10} \rightarrow H $ is a homomorphism. How do I describe and justify all the possible images of $f$. $D_{10} = ({1, \sigma, \sigma^2, \sigma^3, \sigma^4, ...
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0answers
24 views

Why is this group of matrices isomorphic to the dihedral group?

I am reading Abstract Algebra, Theory and Applications by Judson and in exercise $13$ chapter $9$, Isomorphisms, I need to prove that the set of matrices $$A=\pmatrix{ \omega & 0 \\ 0 & \omega ...
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1answer
51 views

About proving that $\operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n)$ [closed]

How can I prove that $$ \operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n), $$ where $\mathbb {D}_n$ is the dihedral group. Can someone help me please? ...
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0answers
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How to find the absolutely irreducible representations of $D_4$ dihedral group

I want to calculate the irreducible representations of $D_4$ and ultimately the absolutely irreducible representations. Right now what I do is since I know the group order is $2n=2(4)=8$ write down ...
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1answer
46 views

Is there a surjective homomorphism from $D_{10}$ to $\mathbb Z _2\times \mathbb Z_2$

Is there a homomorphism $f\colon D_{10} \to \mathbb Z_2\times \mathbb Z_2$ that is onto? Attempt: $D_{10} = \{e,s,r,...,r^9,sr,...,sr^9\}$, $\mathbb Z_2\times \mathbb Z_2 = ...
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2answers
57 views

Show the normal subgroups and cosets of a dihedral group (D6)

$G=D_6$ and $H=<R^2>$. Use this Cayley table for $D_6$ (a). Show that $H \vartriangleleft G$. I want to show by finding out $aH=Ha$ for all $a \in G$, but then how do I proceed, it would be ...
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0answers
51 views

Is $D_{2n}$ isomorphic to $D_n \times \Bbb{Z}_2$ for all $n$? For all odd $n$?

Is $D_{2n}$ isomorphic to $D_n \times \Bbb{Z}_2$ for all $n$? For all odd $n$? I just want to see if my thinking is sound here. My thought process is this. $\mathbb{Z}_2 \cong \{e,j\} \subset ...
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1answer
50 views

Kernel of a homomorphism (dihedral group)

Suppose that $f: D_{18} \to GL(2,\Bbb R)$ is a homomorphism, $\lvert r \rvert = 18$ and $f(r) = R := \begin{pmatrix}1& 1 \\ -1& 0\end{pmatrix}$. What is $\lvert \ker(f) \rvert$? Attempt: I ...
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1answer
40 views

Symmetric Group and Dihedral Group relationship

Show that $D_3 = S_3$ but $D_n \subsetneq S_n$ for $n \geq 4$. In my class, we proved that $D_n$ is generated by $f = (1,2,3,...,n)$ and $g=(1)(2,n)(3,n-1)...(\frac{n+1}{2}, \frac{n+3}{2})$ (for an ...
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1answer
46 views

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 ...
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1answer
35 views

Are dihedral groups well defined by their generating groups?

It is well known that $D_n=<r,s | r^n=s^2=id, srs=r^{-1}>$. Now, given a group of 2n elements and a generating group of two elements satisfying the above relations is it isomorphic to $D_n$?
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0answers
8 views

What exactly is a Frieze group and how would you find the isometries preserving one?

So far, I've come across several examples of frieze groups, but I've not yet come across an understandable definition of what they are. I've also been asked questions that ask me to state the ...
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1answer
11 views

Show the sub group $\langle r^k\rangle$ is normal in the dihedral group of order $2n$ when $k | n$ and $r$ is the rotation by $2\pi/n$

let $ k | n ; n,k \in \Bbb Z $ let $r$ be the rotation in the plane by an angle $2 \pi \over n$ prove the subgroup $ \langle r^k\rangle $ of $D_n$ is normal. Further is there a normal subgroup ...
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1answer
11 views

The order deduced from relations in $D_n$

If $D_n \triangleq \langle a,b | a^n=e, b^2=e, abab=e \rangle$, can it be proved that the order of $a, b$ is actually $n$ and $2$ respectively ? I mean can the relations on the right somehow after ...
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1answer
39 views

$D_n$ is a group for all integers $n\geq 3$

If one wants to prove that $D_n$ (a dihedral structure) is a group for all $n\geq 3$, would it be sufficient to state the following? $D_n = \{1, s, r, ..., r^n, sr, ..., sr^n\}$. Associativity: ...
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1answer
27 views

Suppose $F_1$ and $F_2$ are distinct reflections in $D_n$ such that $F_1F_2=F_2F_1$…

Suppose $F_1$ and $F_2$ are distinct reflections in $D_n$ such that $F_1$$F_2=F_2F_1$, prove that $F_1F_2=R_{180}$. I'm stumped on where to even start. Up to the point where I've gotten the book I am ...
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0answers
25 views

Automorphism group of disjoint cycle graphs of different lengths

This question is supplementary to another question. From that question, we know that the automorphism group of the $N$ disjoint cycle graphs of same length $n$ is $S_N \wr D_n$. My question: What is ...
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0answers
15 views

Automorphism group of disjoint cycle graphs, $N C_n$

I understand that the automorphism group of a cycle graph, $C_n$ is the dihedral group $D_n$ of order $2n$. My question: Let's define $N C_n$ be the disjoint union of $N$ number of $C_n$. What will ...