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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7
votes
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
16k 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
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2answers
283 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 ...
-3
votes
1answer
841 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 ...
15
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1answer
701 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 &...
11
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1answer
455 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
281 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
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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 ...
0
votes
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.
6
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5answers
11k views

Is it true that a dihedral group is nonabelian?

Is it true that a dihedral group is nonabelian? I'm not sure if the result is true. I checked it for some lower order and I think the result may correct. But I failed to prove/disprove the result.
14
votes
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
410 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
987 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 $D_n$. ...
3
votes
2answers
534 views

Prove that the number of subgroups in $D_n = \tau (n) + \sigma (n)$

Prove that the number of subgroups in $D_n = \tau (n) + \sigma (n)$ where $\tau (n)$ represents number of divisors of $n$ and $\sigma (n)$ represnts the sum of divisors of $n$. Attempt: $D_n = \{e,r,...
10
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1answer
1k 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 $2$-...
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
133 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=D_{2n}=⟨x,y|x^2=y^n=e, $ $yx=xy^{n-1}⟩$ I need to find $G'$ [ the commutator of G] now I understand that $G'$ is the subgroup generated from $ U=xyx^{-1}y^{-1} , $ $ \ \forall x,y \in G$ So,...
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 ...
2
votes
2answers
303 views

Conjugacy classes for rotations of $D_{2n}$

It says in my notes that for a dihedral group $D_{2n}$, if $n$ is even, then each conjugacy class has at most $1$ element. It says that for the conjugacy class $C_{r^k}$ where $r^k$ is some reflection,...
3
votes
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
742 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 "...
3
votes
1answer
108 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 $o(\rho)=n$,...
2
votes
3answers
310 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
145 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
558 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
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
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
vote
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
votes
1answer
424 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
5answers
251 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 ...
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_{...
3
votes
1answer
406 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 ...
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 ...
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
106 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
76 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
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
1answer
58 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
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
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
vote
0answers
79 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
478 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
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
93 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? ...
0
votes
0answers
62 views

Terminology for dihedral groups

What notation is most common for the dihedral group of order $2n$? I'm talking about the group of symmetries of a regular $n$-gon. I know that some books call this group $D_n$, and some books call it $...