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
0answers
23 views

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 ...
0
votes
1answer
23 views

Showing that $\phi$ is a homomorphism

$G=\mathbb{Z}^2$ is a group with product $(a,b)\cdot(c,d)=(a+c,(-1)^cb+d)$. Show that the image $\phi: G \to D_{10}$ with $(a,b) \mapsto s^ar^b$ is a homomorphism ($D_{10}$ is the dihedral group of ...
2
votes
3answers
32 views

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 ...
3
votes
3answers
80 views

Can the dihedral groups be seen as subgroups of each other?

I am solving one difficult problem and now I need information on this: If $m\mid n$, is then possible to "embed" $D_m$ to $D_n$, or otherwise said, does $D_n$ have a subgroup isomorphic to $D_m$? ...
0
votes
1answer
58 views

Non-trivial graph automorphism groups with $D_n$ as subgroup

I understand that the automorphism group of an $n$ cycle graph is the dihedral group $D_n$ of order $2 n$. From the comment of @Christian, I also understand that $S_n$ is the automorphism group of the ...
2
votes
2answers
56 views

Dihedral groups: Relationship between symmetries and rigid motions

In Dummit & Foote, $D_{2n},\ n \geqslant 3$ is the set of symmetries of a regular $n$-gon, where a symmetry is a rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-gon, ...
0
votes
1answer
39 views

What does congruency mean in $D_4$?

What does congruency mean in $D_4$? How can I check for example that For $K = \{k_0, k_2\}$, $$p_x \equiv p_y \pmod K$$ I.e. how to evaluate $(p_x - p_y) \bmod K$, specifically what is $(p_x - p_y)...
2
votes
1answer
54 views

Conjugacy classes of $\mathcal D_{10}$.

I was wondering if there is a special technique to find the conjugacy classes of $\mathcal D_{10}=\left<a,b\mid a^5=b^2=1,bab^{-1}=a^{-1}\right>$, and of $\mathcal D_{2n}=\left<a,b\mid a^n=b^...
0
votes
0answers
13 views

Inner product of Induced permutation representation and an irrep $\langle {\chi \uparrow^{S_n}_{D_n}}_{\mathbf{ 1}_{D_n}} , \chi_\rho \rangle_{S_n}$

I am trying to compute the inner product of the characters of the induced permutation representation from the trivial representation of a dihedral group $D_n$ of order $2 n$ to $S_n$ and an irrep $\...
0
votes
2answers
35 views

Finding groups $H$ for which there exists surjective homomorphisms $f:D_4 \rightarrow H$?

How can I find out for which groups $H$ there exists surjective homomorphisms $f: D_4 \rightarrow H$? $D_4$ is the dihedral group of the square. I have a theorem that says that there exists such ...
1
vote
1answer
47 views

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 ...
1
vote
2answers
71 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
46 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
67 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 = \left<...
2
votes
1answer
37 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
22 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 representations ...
9
votes
0answers
82 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
1answer
73 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
23 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
52 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
42 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 $\Rightarrow$...
1
vote
1answer
84 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
26 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: $D_{4}=[1,r,s,sr^{-1}]$,...
3
votes
5answers
105 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
29 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
45 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
28 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 $D_{n/...
2
votes
2answers
54 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
34 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 $$|D_{2n}:N_{D_{2n}}(D_{4})|=|\mathcal{C}|,$$...
1
vote
0answers
24 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
36 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 ...
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 s_j)^{...
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 $N_i$ symbols,...
0
votes
1answer
51 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
25 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
50 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
60 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
287 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
110 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 \...