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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

7
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1answer
5k views

Center of dihedral group

I am trying to solve the following exercise about the dihedral group and its center: If $g\in Z(D_{2n})\Leftrightarrow ga=ag, bg=gb$, where $a,b$ are generators of $D_{2n}$. We have defined the ...
14
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2answers
15k views

Subgroups of $D_4$

I need to determine the subgroups of the dihedral group of order 4, $D_4$. I know that the elements of $D_4$ are $\{1,r,r^2,r^3, s,rs,r^2s,r^3s\}$ But I don't understand how to get the subgroups..
6
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1answer
2k views

Prove that the dihedral group $D_4$ can not be written as a direct product of two groups

I like to know why the dihedral group $D_4$ can't be written as a direct product of two groups. It is a school assignment that I've been trying to solve all day and now I'm more confused then ever, ...
7
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4answers
1k views

Some Subgroup of Dihedral Group is Normal

I ran into this question when I was studying for my abstract algebra midterm. Show that the subgroup $H$ of rotations is normal in the dihedral group $D_n$. Find the quotient group $D_n/H$. I'm ...
4
votes
2answers
259 views

Center of $D_6$ is $\mathbb{Z}_2$

The center of $D_6$ is isomorphic to $\mathbb{Z}_2$. I have that $$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$ $$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$ My ...
4
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3answers
2k views

$\mathrm{Aut}(D_4)$ is isomorphic to $D_4$

Problem statement: I need to find out if $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$ and explain my answer. I already know that it is isomorphic, so now all I need to do is to prove it. I assume that ...
11
votes
1answer
444 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 \...
3
votes
3answers
276 views

Why Composition and Dihedral Group have reverse order of operation?

NOTE - I didn't receive any answer in here and I think because my first post is not clear, so I entirely made another example: $K={\{id,r^2,r^4,s,r^2s,r^4s}\}$ is a proper subgroup of the dihedral ...
0
votes
1answer
5k views

How to describe all normal subgroups of the dihedral group Dn? [duplicate]

The dihedral group consists of rotations and symmetries. But the symmetry group is a group only if n is even, thus the group of rotations is a normal subgroup of the dihedral group. So how to ...
14
votes
1answer
650 views

Group cohomology of dihedral groups

If $m$ is odd, the group cohomology of the dihedral group $D_m$ of order $2m$ is given by $$H^n(D_m;\mathbb{Z}) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/(2m) & n \equiv 0 \bmod 4, ~ n &...
14
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3answers
8k views

Proof that $S_3$ isomorphic to $D_3$

So I'm asked to prove that $$S_{3}\cong D_{3}$$ where $D_3$ is the dihedral group of order $6$. I know how to exhibit the isomorphism and verify every one of the $6^{2}$ pairs, but that seems so long ...
3
votes
1answer
407 views

Prove: $D_{8n} \not\cong D_{4n} \times Z_2$.

Prove $D_{8n} \not\cong D_{4n} \times Z_2$. My trial: I tried to show that $D_{16}$ is not isomorphic to $D_8 \times Z_2$ by making a contradiction as follows: Suppose $D_{4n}$ is isomorphic to $...
3
votes
1answer
965 views

Prove two reflections of lines through the origin generate a dihedral group.

Let $l_1$ and $l_2$ be the lines through the origin in $R^2$ that intersect in an angle π/n and let $r_i$ be the reflection about $l_i$. Prove the $r_1$ and $r_2$ generate a dihedral group $D_n$. ...
5
votes
3answers
1k views

On the centres of the dihedral groups

In an proof that I recently read, the following 'fact' is used, where $D_{2n}$ denotes the dihedral group of order $2n$: If $n$ is even, then $D_{2n} \cong C_2 \times D_n$. The (short) given ...
2
votes
3answers
131 views

$D_3\oplus D_4$ not isomorphic to $D_{24}$

We need to prove that $D_{3} \oplus D_{4}$ is not isomorphic to $D_{24}$ . The way in which I approach such type of questions is to count the number of elements of order $x$ in one group and then in ...
0
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1answer
3k views

understanding the commutator of dihedral group [duplicate]

Let $G=D2n=⟨x,y|x^2=y^n=e, $ $yx=xy^{n-1}⟩$ i need to find G' [ the commutattor of G] now i understand the G' is the subgroup that is generated from $ U=xyx^{-1}y^{-1} , $ $\forall x,y \in G$ so ...
9
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1answer
979 views

Finding Sylow 2-subgroups of the dihedral group $D_n$

I am trying to describe the Sylow $2$-subgroups of an arbitrary dihedral group $D_n$ of order $2n$. In the case that $n$ is odd, $2$ is the highest power dividing $2n$, so that all Sylow $2$-...
3
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1answer
2k 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 ...
0
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1answer
1k 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
822 views

A group is generated by two elements of order $2$ is infinite and non-abelian

My question is as follows: Let $G = \langle a,b \mid a^2=b^2=1 \rangle $ be a group generated elements $a, b$ and the equation $a^2=b^2=1$. Prove that $G$ is infinite and non-abelian. I got the ...
3
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2answers
2k views

Algebra - Infinite Dihedral Group

Let $G$ be the set of bijections $\mathbb{R} \to \mathbb{R}$ which preserve the distance between pairs of points, and send integers to integers. Then $G$ is a group under composition of functions. The ...
1
vote
2answers
3k 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 ...
3
votes
1answer
712 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 "...
2
votes
3answers
307 views

Intuition - $fr = r^{-1}f$ for Dihedral Groups - Carter p. 75

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square's vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles's $f$ is different. Carter fleshes out why $...
2
votes
1answer
124 views

Group isomorphism between $D_3$ and $S_3$

If one wants to prove that $D_3$ is isomorphic to $S_3$, would it be sufficient to define a homomorphism $\psi: D_3\to S_3$ and argue that it is well-defined since $\psi(sr^i)=\psi(s)\psi(r)^i=\...
2
votes
1answer
531 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^...
1
vote
1answer
78 views

Find the inner and outermorphisms of a particular dihedral group

Given that |Inn($D_8$)| = 8 and |Out($D_8$)| = 2 where Out($D_8$) = Aut($D_8$)/Inn($D_8$) and $D_8$ = {e,r,$r^2$,..,$r^7$,s,sr,...,$sr^7$} we want to find Inn($D_8$) and Out($D_8$). We know that Out(...
1
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2answers
83 views

Find $G/Z(G)$ given the following information about the group?

$G$ is a finite group generated by two elements $a$ and $b$, we are given the following data: Order of a= $2$ Order of $b=2$ Order of $ab=8$. If $Z(G)$ denotes the center then what is $G/Z(G)$ ...
1
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0answers
81 views

Questions about the dihedral group $D_8$ [duplicate]

Consider the dihedral group $D_8$ of order $16$. Consider $D_8$ with the presentation $D_8=\{r^i s^j : i=0,...,7; j=0,1; r^8=s^2=e; sr=r^7s=r^{-1}s\}$, where $\{e\}, \{rs, r^3s, r^5 s, r^7s\}$ and $\{...
5
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1answer
423 views

Could someone explain chirality from a group theory point of view?

While answering this question my interest in the rotation/reflection group was piqued. I personally know very basic group theory, not much more than what a group really is. I understand that the ...
4
votes
1answer
1k views

Composition Series for Dihedral Groups

I have been thinking about a composition series for $D_{14}\times D_{10}$ (where $D_{2n}$ is the dihedral group with $2n$ elements). Is the following a correct composition series for $D_{10}\times D_{...
4
votes
5answers
247 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 ...
3
votes
2answers
76 views

Dihedral subgroups of $S_4$

Prove that in $S_4$ there are $3$ groups that are isomorphic to $D_4$. I know that the $2$-sylows of $S_4$ should be subgroups of order $8$, but to prove it is a bit tricky for me Any help would be ...
3
votes
1answer
105 views

Prove that $| \operatorname{Aut}(D_n)|\le n\phi(n)$

Prove that for $n\gt 2$, $| \operatorname{Aut}(D_n)|\le n\,\phi(n)$ where $D_n$ is the dihedral group with 2n elements and $\phi$ is Euler phi function. Let $\rho$ be a rotation such that $o(\rho)=n$,...
3
votes
1answer
396 views

Symmetry group on integers

Construct a symmetry group for the set of integers on the number line that generalizes the dihedral group to have a countably infinite, rather than finite size. Treat the integers as vertices. What ...
2
votes
3answers
86 views

Showing an Isomorphism between question group of $S_4$ and $D_6$

I have a subgroup $N$ of $S_4$, where $ N = [1, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)] $ I need to explain whether quotient group $G/N$ is isomoprhic to either $C_6$ or $D_6$ (no proof required, just an ...
2
votes
1answer
103 views

Find all homomorphisms from $D_{2n}$ to $\mathbb C^\times$ (revisit)

I actually was asking the same question in here but haven't gotten any feedback yet. I now can elaborate a little so that final answer would be closer. I wanted to find all homomorphisms from the ...
2
votes
1answer
75 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? ...
1
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1answer
91 views

How to geometrically show that there are $3$ $D_4$ subgroups in $S_4$?

As shown in this note, the symmetry group $S_4$ for a cube has $3$ subgroups that are isomorphic to $D_4$, the dihedral group of order $2 \times 4 = 8$. How to geometrically illustrate this fact? ...
1
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1answer
52 views

Coloring sides of truncated triangular dihedral(bipiramid) into 3 colours

I need to find out the amount of ways to colour truncated triangular dihedron into 3 colours. So, the task will be easier if I had simple triangular dihedron. First of all, do I understand right ...
1
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0answers
77 views

|${G_x}$| of ${D_{10}}$

I am looking for the order of the stabilizer group of $D_{10}$. I know that ${G_x} = \{g \in G : gx = x\}$. I am curious what to use for $x$ though? Should I just cycle through elements of $D_{10}$ ...
1
vote
1answer
149 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 ...
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0answers
62 views

Irreducible representation of $C^*(D_\infty)$

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$ Ultimately, I'm interested in finding a ...
1
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1answer
449 views

Automorphisms in the Dihedral groups

Let g be a group and $a \in G$. Define $\phi_a:G\rightarrow G$ by $\phi_a(g)=aga^{-1}.$ Now Let $G=D_4$ and $a=r$, where $r$is the rotation. We must show that $\phi_r: D_4\rightarrow D_4$. So show ...
1
vote
1answer
213 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/...
1
vote
1answer
90 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?
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3answers
363 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 ...
0
votes
0answers
78 views

How to show $D_3\oplus D_4$ is not isomorphic to $D_{24}$? [duplicate]

How to show $D_3\oplus D_4$ is not isomorphic to $D_{24}$? Here $D_n$ is the dihedral group of order $2n$. I am not sure how to prove this. I am not very good with the dihedreal groups.
0
votes
0answers
41 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 ...
0
votes
0answers
623 views

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

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. $\mathbb{Z}_2 \cong \{e,j\} \subset D_n$ ...