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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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Showing reflections in midpoint of sides are not equal to rotations

I want to prove that the order of $D_{2n}$ is equal to $2n$. I said that there is $n$ distinct rotations ,so there must be $n$ reflection. when $n$ is odd where n is the number of sides of regular n-...
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Does there exist a group $G$ such that $\operatorname{Aut}(G)\cong D_5$, where $D_5$ denotes the dihedral group of order 10?

I came across this problem stated in the title having no clue what to do, and got stuck even in the finite case. Here's my attempt: I first proved that the group $G$, if it exists, cannot be finite ...
Cyankite's user avatar
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Dihedral Groups as Coxeter Groups

I have just completed Exercise 1.2 in the book "Combinatorics of Coxeter Groups" stated below: Show that there exist Coxeter systems $(W,S)$ and $(W',S')$ with $|S|\neq|S'|$ such that $W\...
capoocapoo's user avatar
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1 answer
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Automorphism of order 2 of $D_q$

Let $q\geq 3$ be a prime number and $D_q$ be the Dihedral Group of order $2q$. Find all automorphism of $D_q$ of order $2$. I tried this using a 'generic' automorphism $\varphi$ such that $\varphi(r)=...
Thomas García Villar's user avatar
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3 answers
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Dihedral group $D_3$(also $D_4$) - reflection compose rotation

I’ve read in a book(image link)that for dihedral group $D_3, a \circ r_1 = b $where a means reflection about $AO$ ($O$ is the centroid of an equilateral triangle $ABC$), $r_1$ means rotation about O ...
A Ghosh 's user avatar
3 votes
1 answer
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Irreducible $2$-dimensional representation of $D_n$

Show that for $n\geq3$ the dihedral group $D_n$ has an irreducible representation of dimension $2$. Is it unique? I thought of doing it this way: let's count the conjugacy classes of $D_n$, which are, ...
Andreadel1988's user avatar
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Finding irreducible representations of $D_{2n}$ using Mackey little group method

Let $D_{2n}$ be the dihedral group on 2n elements, consisting of n rotations and n reflections. I know the group of n rotations form a normal subgroup of $D_{2n}$ and $D_{2n}$ is a semidirect product ...
mathlover's user avatar
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2 answers
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Galois Group of $x^n-p = D_n$

The question I have on my homework is to prove the following theorem. If $n \in \{3, 4, 6\}$ and $p \in \mathbb{Z}_{>0}$ is prime, then the Galois group of $x^n − p$ is the dihedral group $D_n$. ...
Nic's user avatar
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What are the groups containing dihedral group $D_4$ of order $8$? [closed]

I'm a little embarrassed to ask this but I couldn't answer it myself. I am looking the groups that contains $D_4$ and larger than $D_4$. Here is what I think: We cannot say $D_4 \subseteq D_n, n\ge 5$ ...
Elise9's user avatar
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How to prove the enhanced power graph of dihedral groups?

According to Aalipour et al. (2016), they defined the enhanced power graph of group $G$ as a simple undirected graph where the vertices are all elements of $G$ and two vertices, $x$ and $y$, are ...
nlydamohd's user avatar
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1 answer
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How can we distinguish elements of $D_n$ that include reflections versus those that don't?

For $n \geq 3$, the dihedral group $D_n =\langle r, s \rangle$, where $r ^n = s^2 = e$. Within this group, we can distinguish two types of elements: Those of the form $r^i$, where $i$ is any integer ...
SRobertJames's user avatar
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Proving the isomorphism type of the commutator subgroup of the dihedral group $D_n$

For any $n$, to what group is the commutator subgroup of the dihedral group $D_n$ isomorphic to? My solution is below. I request verification, feedback, and improvements. In particular, can you help ...
SRobertJames's user avatar
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Counting subgroups of dihedral groups

Question: How many subgroups of order 6 does $D_6$ have? How many does $D_{12}$ have? Generalize to $D_n$ where n is a positive integer divisible by 6. Here, order of $D_n = 2n$. Please don't use ...
Learner's user avatar
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How does $ab= a^2ba$ for the Dihedral group $D_8 = \langle a, b \rangle$ and a $\mathbb{C}D_8$-Algebra?

I'm trying to show for $z = b + a^2b \in \mathbb{C}D_8$, $az = za$. I have the presentation $D_8 = \langle a, b \ : \ a^4 = 1, b^2 = 1, b^{-1}ab = a^{-1} \rangle$. So I want to show $az = ab + a^3b = ...
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Arithmetic on elements of $D_{2n}$

I've seen the arithmetic on the dihedral group $D_{2n}$ written in several different ways, but here's what I'm working with. Say that $r$ is a clockwise rotation by $\frac{2\pi}{n}$ radians and $s$ a ...
Cardinality's user avatar
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Supposed contradiction from Sylow theorems

Let $D_{506}$ denote the dihedral group with $1012 = 4 \cdot 23 \cdot 11$ elements. The Sylow theorems tell us that the number of 2-Sylow-subgroups (that is, subgroups of order 4) of $D_{506}$ divides ...
univalence's user avatar
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Basic Group Theory Question

Consider the dihedral group $D_3$ of order $|D_3|=6$ comprised of the $6$ symmetries $D_3=\{e,r,r^2,rs,r^2s\}$ of an equilateral triangle ($e=r^0s^0$ is the do-nothing symmetry, $r$ is a rotation by $...
William Deng's user avatar
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Automorphisms of the Infinite Dihedral Group

Let $D_\infty$ be generated by $a$ the reflection fixing the line $\{x=0\}$ and $b$ the reflection fixing the line $\{x=1\}$ in $\mathbb{R}^2$. I want to show that any $\alpha\in \text{Aut}(D_\infty)$ ...
mathemagician99's user avatar
2 votes
2 answers
249 views

A question regarding the dihedral group $D_4$.

I know that the dihedral group $D_4$ is generated by two elements $\sigma$ and $\tau$ such that $\sigma^4=\tau^2=e$, and $\tau\sigma\tau^{-1}=\sigma^3$. This is essentially the group of eight elements ...
Aravind Gundakaram's user avatar
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Finding a non-abelian group with $54$ elements non-isomorphic to $D_{27}$

My goal is to find a non-abelian group with $54$ elements non-isomorphic to $D_{27}$ and for that matter, I tried $\mathbb{Z_3} \times D_9$, now we have that $Z(\mathbb{Z_3} \times D_9) =\mathbb{Z_3} \...
J P's user avatar
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Compute normalizer of $\left<s,r^4\right>$ in $D_{16}$

This is from a problem in Dummit&Foote. Here is my attempt: $$\left<s,r^4\right> = \{1,s,r^4,r^8,r^{12},sr^4,sr^8,sr^{12}\}$$ We only need to check if the conjugate of a generator is an ...
user108580's user avatar
1 vote
1 answer
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Method for classifying irreducible $\mathbb{C}$-representations of $D_{10}$ of dimension $2$

$D_{2n} = \{ r, s: r^n = s^2 = e, srs = r^{-1} \}$ is the dihedral group with order $2n$. I'm trying to classify the $2$-dimensional irreducible $\mathbb{C}$-representations of $D_{10}$ up to ...
Robin's user avatar
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Normalizer of Sylow $2$-subgroup of $D_{2n}$.

In Normalizer of a Sylow 2-subgroups of dihedral groups it is proved that the number of Sylow $2$-subgroups of $D_{2n}$ if $2n=2^ak$ where $k$ is odd is $k$ without proving $N_G(P)=P$ if $P$ is Sylow $...
Laurence PW's user avatar
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$H,K \leq G$, $|H| = 126, |K| = 228$. Show that $H \cap K \leq G$ is either cyclic or isomorphic to $D_3$

So, I don't have to prove that $H \cap K$ is a subgroup of $G$, and $G$ is finite. I know that any group of order $6$ is isomorphic to either $D_3$ or $\mathbb{Z}_6$ (which is cyclic) so I'm assuming ...
spooleey's user avatar
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1 answer
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Binary sequences of $a$ zeros and $b$ ones which have $a+b$ circular shifts.

Let $P(a;b)$ be the set of all sequences generated by permutation of $a$ zeroes and $b$ ones. $$|P(a;b)|=\binom{a+b}{a}$$ I want to prove that if $gcd(a,b)=1$, every sequence in $P(a;b)$ will have $a+...
Nothing special's user avatar
1 vote
1 answer
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Proving a non-cyclic $G$ has a cyclic subgroup of index $2$ and $G\cong D_k.$

Suppose $G$ is a non-cyclic group generated by two elements, both of order $2.$ Prove that $G$ has a cyclic subgroup of index $[G:C]=2.$ If such a group is of finite order $2k,$ prove it is isomorphic ...
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Listing all subgroups of $D_{10}$

My work so far: The cyclic subgroups are trivial to find. They are simply rotations generated from $\langle r^d\rangle$ where $d\mid10$. So $d=1,2,5,10$. $\langle r \rangle= \{e,r, r^2, \dots, r^9 \}$ ...
Peetrius's user avatar
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1 vote
1 answer
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Showing that $f(r^k)^{-1}=f(r^k)$, where $f$ is a representation of the dihedral group $D_n$

Let $D_n$ be the dihedral group, which contains elements of the form $\{r^k\}_{k\in K}$ and $\{sr^k\}_{k\in K}$ where $K=\{1,\dots,n-1\}$ and $r$ and $s$ satisfy the equations $$ r^n=s^2=1,\quad sr^ks=...
Logi's user avatar
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0 answers
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Finding orbit in a group action of automorphism group of dihedral group on the dihedral group.

Finding orbit in a group action of automorphism group of dihedral group on the dihedral group. Let $$D_{2n}=\langle a,b:a^n=b^2=1,bab^{-1}=a^{-1}\rangle $$ be the dihedral group of order $2n$ and $...
idiot's user avatar
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1 vote
2 answers
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No subgroup of $D_3$ has order 4.

Question: Prove that the dihedral group of order $6$ has no subgroup of order $4$. I am trying to prove above question. But i have no idea how can i start! Just before this question (in exercise) i ...
Afzal's user avatar
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3 votes
1 answer
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If $|G|=154$, and it not a direct product, then it must be isomorphic to $D_{154}$.

Suppose $|G|=154$ and $G$ is not decomposable in the sense that there are no proper subgroups $H$ and $K$ of $G$ such that $G \cong H \times K$. Show that $G \cong D_{154}$. Here is my attempt: Let $P ...
John Smith's user avatar
1 vote
1 answer
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Subgroup generated by two elements of group.

In the book Contemporary Abstract Algebra by Gallian. Chapter 3 (edition 9). Exercise 3 says What can you say about a subgroup of $D_3$ that contains $R_{240}$ and a reflection $F$? What can you say ...
Afzal's user avatar
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2 votes
2 answers
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Smallest $m$ such that $D_n$ is a subgroup of $S_m$

I was wondering what was, given any $n$ the smallest $m$ such that $D_n$ is a subgroup of $S_m$. For cycle groups $C_n$ I believe there is a relation of the form $m = \min \sum_{k=1}^{K}p_k$ with : $...
Pierrick Leroy's user avatar
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What are the elements of the Dihedral Group $D_8$ when given by $D_8 = \langle a, b \mid a^2 = b^2 = (ab)^4=1\rangle?$ [duplicate]

I'm very confused since I've always seen the Dihedral Group $D_8$ given as $$D_8 = \langle a, b \mid a^4 = b^2 =1, ab= a^3b\rangle.$$ I'm given the Definition $$D_8 = \langle a, b \mid a^2 = b^2 = (ab)...
tomato's user avatar
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0 answers
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Symmetries of the dihedral groups $D_n$ and $D_{nh}$.

According to my lecture book (Linear Representations of Finite Groups by Serre), the dihedral group $D_n$ consists of $n$ rotations and $n$ reflections in the plane that preserve a regular polygon ...
Jacob's user avatar
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Proving that $r^i \neq sr^j$ in $D_{2n}$ [duplicate]

I was trying to proof that the order of the dihedral group $D_{2n}$ is $2n$ given its presentation $$ D_{2n} = \langle r,s ~\vert~ r^n = s^2 = 1 ,~ rs = sr^{-1} \rangle $$ and I further assumed that ...
Hesham Abdelgawad's user avatar
1 vote
0 answers
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Importance of abelian subgroups of dihedral group in computing conjugacy classes?

Let $D_{8}$ be the dihedral group of order 8. We would like to find its conjugacy classes. Authors Dummit and Foote write: In $D_{8}$ we may also use the fact that the three subgroups of index 2 are ...
Jaspreet's user avatar
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3 votes
1 answer
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Unique $p$-Sylow subgroup of Dihedral group $D_n$.

If $p>2$ is a prime number that divides $n \in \mathbb N$, then $D_n = \langle r,s \rangle$ (with $r^n=s^2=id, sr=r^{-1}s$) has a unique $p$-Sylow subgroup $H \leq D_n$. I'm having trouble with ...
RatherAmusing's user avatar
3 votes
0 answers
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Embedding $D_{2n}$ in $U_2$

I have a problem doing Exercise 18.8 of groups and symmetry by M. A. Armstrong. 18.8. Look at our description of the Möbius band as a subset of $\mathbb{C} \times \mathbb{C}$ and find matrices in $...
hbghlyj's user avatar
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2 votes
0 answers
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When does $|G|=pq$ admit unique non abelian representation? [duplicate]

Classifying $G$ with $|G|=pq$ has been asked several times in this site, but I am having trouble with number theory. Let $q<p$, $Q=\langle u\rangle\in \text{Syl}_q(G)$, $P=\langle v\rangle\in \text{...
Kadmos's user avatar
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1 vote
1 answer
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Representation of symmetry group $D_4$ acting on a square

Consider the square with vertices $(\pm 1, \pm 1)$. The symmetry group $D_4$ of this square acts by permuting the vertices Show that each permutation comes from a linear transformation. Compute the ...
Sayan Dutta's user avatar
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Let $n=dm$ and consider the group $D_n$ , with $K=⟨ρ^m⟩$ .Show that the following $2m$ cosets of $K$ are all distinct:

$D_n = \langle \mu, \rho \mid \mu^2 = \rho^n = \epsilon, \mu\rho = \rho^{-1}\mu \rangle$ Let $n=dm$ and consider the group $D_n$, with $K = \langle \rho^m \rangle$. Show that the following $2m$ cosets ...
user20194358's user avatar
1 vote
1 answer
45 views

How do i show that $\psi : D_n \rightarrow D_m$ defined by $\psi(\mu^i \rho^j) = \delta^i \gamma^j$ is a homormophism from $D_n$ to $D_m$ where n =qm

Let $n = dm$ Consider these groups: $D_n = \langle \mu, \rho \mid \mu^2 = \rho^n = \epsilon, \mu\rho = \rho^{-1}\mu \rangle$ $D_m = \langle \delta, \gamma \mid \delta^2 = \gamma^m = \epsilon, \delta\...
user20194358's user avatar
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1 answer
309 views

Definition and Properties of Dihedral Group

Definition: Here is equivalent definition of dihedral group $$\begin{align}D_{2n}&=\{\sigma \in S_n\mid \sigma(a+1)\equiv\sigma(a)+1\pmod n, \forall 1\leq a\leq n\} \bigcup \{\sigma \in S_n\mid \...
user264745's user avatar
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0 votes
2 answers
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Help understanding how to show that two groups are isomorphic

I'm trying to understand what is required to show that two groups are isomorphic. My understanding is that two groups $G_1$ and $G_2$ are isomorphic if: $ |G_1| = |G_2| $ (equal cardinality) $G_1$ ...
user1160465's user avatar
1 vote
1 answer
292 views

Definition of Dihedral Group in Dummit’s Abstract Algebra

For each $n\in \Bbb{Z}^+$, $n\geq 3$ let $D_{2n}$ be the set of symmetries of a regular $n$-gon, where a symmetry is any rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-...
user264745's user avatar
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4 votes
0 answers
114 views

Geometrical interpretation of the faithful actions of $D_6$ and $D_{10}$ on sets of size $5$ and $7$, respectively. [duplicate]

The dihedral group of order $2n$, $D_n$, acts faithfully on the set of the $n$ vertices of the regular $n$-gon. For $n=6$ and $n=10$, $D_n$ acts faithfully also on sets of size smaller than $n$, ...
Kan't's user avatar
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2 votes
3 answers
75 views

The dihedral group $D_n$ cannot be written as a product of two nontrivial groups

Let $n$ be an odd prime. I want to prove that the dihedral group $D_n$ cannot be written as a product of two nontrivial groups. But it seems problematic, as I completely neglect the properties of ...
user avatar
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0 answers
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Understanding cycle notation for a dihedral group and one of its subgroups

I'm looking at an example that instructs me to consider the group: $$ G = \langle(1, 2, 3, 4, 5, 6, 7, 8), (1, 8)(2, 7)(3, 6)(4, 5)\rangle \cong Dih(16) $$ And the subgroup: $$ H = \langle(1, 3, 5, 7)(...
lswift's user avatar
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3 votes
0 answers
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Isomorphisms between subgroups of the Symmetry group of the cube

I need to list all the subgroups of the symmetry group of the cube that are isomorphic to either $\mathbb{Z}_n$ or $\mathbb{D}_n$ (the dihedral group) for some $n \in \mathbb{N}$. I know that the ...
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