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

learn more… | top users | synonyms

1
vote
2answers
36 views

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 ...
0
votes
0answers
37 views

Computing the characters of $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$

How can I compute the characters of the induced representation $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$? Here, $S_n$ is the symmetric group over $n$ symbols and $D_n$ is the dihedral group of order $2 ...
2
votes
5answers
58 views

Proving that the groups $S_3$ and $D_6$ are isomorphic [duplicate]

In particular, $S_3$ is the group of permutations of $\{1,2,3\}$, and $D_6$ is the dihedral group of symmetries of the triangle (written as $D_{2\cdot 3}$). In generator-relation form, $D_6 = ...
2
votes
1answer
33 views

Finding normal subgroups

Let $G=\langle e, r,..., r^{n-1},s,sr,...,sr^{n-1} \rangle $ be a dihedral group with $2n$ elements, for $ 3 \leq n$. Prove that the only normal subgroups of $G$ are $\langle r^d \rangle$ (where $d$ ...
0
votes
0answers
21 views

Row in the character table of $D_{10}$

Give the values of one row of the character table of $D_{10}$ corresponding to a character of degree $2$ I know the conjugacy classes of $D_{10}$, the dimensions of the irreducible ...
9
votes
0answers
76 views

Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
1
vote
0answers
60 views

When can a dihedral group $D_{n}$ of order $2 n$ be a $p$-group?

A $p$-group is a group where the order of every group element is a power of the prime $p$. The presentation of a dihedral group $D_n$ of order $2 n$ is as follows. $$D_n = \langle x, y \mid x^n = y^2 ...
0
votes
0answers
22 views

Cycle structure of the generators of the dihedral group

Would the following be correct about generating the dihedral group $D_n$ by permutations? If $n$ is even, the group can be generated as $\langle(2\quad n)(3 \quad n-1) \ldots (\frac{n}{2}-1 \quad ...
0
votes
2answers
47 views

Burnside Lemma and colorings of a $C_{8}$ graph

I'm trying to determine the number of different colorings of the vertices of a cycle $C_{8}$ graph. Suppose I have 10 colors and I suppose I can use every color as much as I want. I consider two ...
2
votes
1answer
41 views

Dihedral group $D_n$ is nilpotent iff $n=2^i$

I want to show that the dihedral group $D_n$ is nilpotent if and only if $n=2^i$ for some $i$. I have shown the direction $\Leftarrow$. Could you give me some hints for the direction ...
1
vote
1answer
56 views

find the center of D8, D10 and Dn [duplicate]

The center of a group $G$ is $$Z(G) = \{g \in G\ :\ \forall x\in G,\ gx = xg\}$$ Find the center of D8. What about the center of D10? What is the center of Dn? I am unsure where to start for this ...
1
vote
1answer
25 views

Is $D_{2n}$ always a subset of $D_{2k}$?

Is $D_{2n}$ always a subset of $D_{2k}$ where $n,k \in N$ and $k>n$ I though this was true as every element of $D_{2n}$ must surely be in $D_{2k}$. For example, consider: ...
3
votes
5answers
102 views

Fastest way to show that $D_6 \to S_5$ is an injective homomorphism

I want to show that there is an injective homomorphism from $D_6 \to S_5$ where $D_6$ denotes the dihidral group of order 12 and $S_5$ the symmetric group. But I'm not sure how I can do this ...
0
votes
0answers
28 views

Find the normal and the composition series

Could you give me some hints how we could find a normal series and all the composition series of $D_4$ ? $$$$ A normal series of $G$ is $$G\geq G\geq G^{(1)} \geq G^{(2)} \geq G^{(3)} \geq \dots ...
0
votes
1answer
16 views

Surjective mapping of matrices under rotational and reflection symmetries

Let me preface this by saying that I'm not a mathematician and that I'm having a hard time stating my problem in the proper terms. Nevertheless, I'm faced with a problem for which I think an elegant ...
1
vote
3answers
44 views

Normal subgroup test

Hi there I have this problem: Is $ <p^6\epsilon^5> $ a normal subgroup of the Dihedral group $ D_4 = \{ I,p,p^2,p^3,\epsilon, p\epsilon, p^2\epsilon,p^3\epsilon \} $? Since I'm not that good at ...
2
votes
0answers
27 views

Dihedral groups acting on Riemann surfaces

I'm studying the quotient riemann surface $X/G$. I'm looking for examples of dihedral groups $D_n$ acting on some riemann surfaces $X$ or at least acting on it's Jacobian JX. Does anybody knows some ...
1
vote
1answer
37 views

How many subgroups of order $n$ does $D_n$ have?

How many subgroups of order $n$ does $D_n$ have? My work: Since subgroups of $D_n$ are either cyclic or dihedral, the subgroups of order $n$ of $D_n$ are $\left< r \right>$ (cyclic) and ...
2
votes
2answers
51 views

How can we find all the subgroups?

I want to find all the normal subgroups of $D_n$. We have that $K$ is a normal subgroup of $D_n$ iff $$gkg^{-1}=k\in K, \forall g\in D_n \text{ and } \forall k\in K$$ right? Could you give me some ...
0
votes
0answers
31 views

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? ...
1
vote
1answer
35 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 ...
1
vote
1answer
53 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 $D_{15}$ ...
1
vote
1answer
22 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 ...
1
vote
0answers
23 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 ...
0
votes
1answer
27 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 KlavĂ­k et al. The notation $\Psi$ was first used expressing ...
0
votes
0answers
9 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, $D_{m_i}$ ...
1
vote
0answers
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 ...
0
votes
0answers
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 $N_i$ ...
0
votes
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 ...
0
votes
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 $N_i$ ...
0
votes
1answer
49 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 ...
0
votes
0answers
13 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 ...
0
votes
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 ...
0
votes
2answers
24 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?
1
vote
1answer
47 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 ...
0
votes
1answer
13 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 ...
-1
votes
1answer
56 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.
4
votes
1answer
280 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 ...
8
votes
1answer
105 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 ...
1
vote
2answers
60 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 ...
2
votes
2answers
43 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, ...
0
votes
1answer
42 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 ...
0
votes
0answers
18 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, ...
0
votes
0answers
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 $$\{S_{N_1} \wr D_{m_1}, \ldots, S_{N_i} \wr D_{m_i}, \ldots, S_{N_t} \wr D_{m_t}\}$$ ...
3
votes
0answers
67 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 ...
6
votes
2answers
39 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 ...
1
vote
1answer
75 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 ...
1
vote
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 ...
3
votes
3answers
48 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 ...
0
votes
0answers
31 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 ...