Questions tagged [dihedral-groups]

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

Filter by
Sorted by
Tagged with
1
vote
1answer
345 views

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 ...
1
vote
1answer
86 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 ...
1
vote
3answers
152 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 ...
1
vote
1answer
210 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. ...
1
vote
1answer
47 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: $(1s)...
1
vote
1answer
204 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 ...
1
vote
2answers
517 views

Generators of a Dihedral Group

Let $D_4=\{ 1,r,r^2,r^3,s, sr, sr^2, sr^3\}$. I want to show that $<s> $ is a normal subgroup of $<s,r^2>$ but $<s>$ is not a normal subgroup of $D_{4}$. I think $<s> = \{1,s\}...
1
vote
1answer
79 views

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 $C$ ...
1
vote
1answer
155 views

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 ...
1
vote
1answer
87 views

Find a composition series of $D_8$

Answer: Let $D_8=\langle a,b\rangle=\{e,a,a^2,a^3,b,ba,ba^2,ba^3\}$, where $a^4 = b^2 = e$ and $ab = ba^{-1}$ as usual. Then $$\{e\}\lhd\langle a^2\rangle\lhd\langle a\rangle\lhd D_8$$ or $$\{e\}\...
1
vote
1answer
322 views

Find all the $p$-Sylow subgroups of $D_6$.

$|D_6|=12=2^23.$ I started with $3$. I know that the number of $3$-Sylow subgroups, denoted $n_3$, is: $1,4,7...$ and I also know that $n_3|2^2$. e.g, $n_3=1, 4$. How can I show that it can't be $4$? ...
1
vote
1answer
425 views

Show isomorphy by mapping generators onto generators only

If I want to show that two cyclic groups are isomorphic, is it enough to show that their cardinality is the same and that the generators of the groups are mapped onto each other? To be precise: I am ...
1
vote
1answer
218 views

Dihedral group of a square $D_4$

Prove that in the $D_4$ a square's symmetry group each element can be uniquely written as $r^i s^j$, $i =1,2,3, \ \ j=0,1$, where $r$ is a rotation by $\frac{\pi}{2}$ around the centre of the square, ...
1
vote
0answers
37 views

Images of the generators of $D_{10}$ under its automorphisms.

I have constructed the dihedral group generated by $a$ and $b$ of order $10$ in GAP by the following way: ...
1
vote
0answers
21 views

What is the name for this generalization of dihedral groups?

A dihedral group $D$ can be defined as the group generated by elements $r$ and $s$, where $r$ has order $n$, $s$ has order $2$, and $sas = a^{-1}$ for all $a \in \langle r \rangle$. It seems that more ...
1
vote
0answers
54 views

Two questions about the dihedral group

First question: 1) Is the sum of subgroup indices of dihedral group with $2n$ elements equal to $\sigma_2(n)+2\cdot \sigma(n)$? Second question: 2) Is $\sigma_2(n)+2\cdot \sigma(n) \le L(H(D_n))$? ...
1
vote
0answers
57 views

How can I show that $D_{2n} \cong C_n \rtimes C_2 $

Let $D_8 := \langle a,b \mid a^4 = 1 = b^2, bab = a^{-1}\rangle$ I'm trying to formally show that $$D_{8} \cong C_4 \rtimes C_2 = \langle s\rangle \rtimes \langle t \rangle$$ My book gives as hint ...
1
vote
2answers
32 views

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}...
1
vote
0answers
34 views

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 ...
1
vote
1answer
20 views

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\...
1
vote
0answers
132 views

Prove or disprove: If $H$ is normal in $G$ and $H$ and $G/H$ are abelian, then $G$ is abelian. [duplicate]

My counterexample I wanted to try is $D_{8}$ and its center $Z(D_{8})$. However, is $Z(D_{8})$ abelian? I am guessing it is because it is cyclic? I am not very knowledgeable with dihedral groups.
1
vote
1answer
97 views

What is the center of $D_{2n}/Z(D_{2n})$

What is the center of $D_{2n}/Z(D_{2n})$. I see that when $n=2^k$ then I have a $p$-group so the center is not trivial. But when $n$ is not power of $2$ how can i know what is the center of this group?...
1
vote
0answers
41 views

Showing a map is well defined (Mobius map to $D_{2n}$ in group theory)

I am asked to show that the subgroup $G$ of the Mobius group $M$, generated by $f(z)=e^{2\pi I/n}z$ and $g(z)=\frac{1}{z}$ is isomorphic to $D_{2n}$. I considered the mapping $$h: G \to D_{2n} \text{...
1
vote
0answers
53 views

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 ...
1
vote
1answer
547 views

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 ...
1
vote
0answers
169 views

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 ...
1
vote
0answers
76 views

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 ...
1
vote
3answers
387 views

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 ...
1
vote
0answers
55 views

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 ...
1
vote
0answers
24 views

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 ...
1
vote
0answers
19 views

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 $N_i$ ...
1
vote
0answers
81 views

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 ...
1
vote
1answer
300 views

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 $D_n$, ...
1
vote
0answers
35 views

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 ...
1
vote
0answers
26 views

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 ...
1
vote
1answer
32 views

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 ...
1
vote
1answer
665 views

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 $r_2$ ...
1
vote
0answers
85 views

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)...
1
vote
0answers
166 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 ...
1
vote
3answers
489 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 ...
1
vote
1answer
89 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
526 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 ...
1
vote
1answer
95 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
48 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
275 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
39 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
41 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 ...
1
vote
0answers
9 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 ...
1
vote
0answers
19 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 \mathbb{Z}_2 =...
1
vote
1answer
83 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 ...