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

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Why are automorphisms of $D_{2n}, n \geq 5$ odd, not always inner?

The dihedral group of order $2n$ is often presented as $$D_{2n}= \langle r,s: r^n=s^2=1, rs= sr^{-1} \rangle \text{,}$$ where $r$ denote rotations and $s$ denote reflections of a regular $n$-gon. ...
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1answer
28 views

Dihedral group of order 2n

I would appreciate if someone could prove this for me: Let G be a dihedral group of order 2n and suppose H is a cyclic quotient group of G. Show that |H|is less than or equal 2.
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1answer
23 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 ...
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1answer
42 views

Find all $x$ in $D_4$ such that $ax(a^{-1}) = b$.

Consider the dihedral group $D_4$. Consider also the elements $a= r_1$ and $b= S_1$ of $D_4$. Find all $x$ in $D_4$ such that $ax(a^{-1}) = b$. Do both $a$ and ($a^{-1}$) cancel each other out? If ...
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1answer
30 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 $4$? ...
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1answer
82 views

Character Table Dihedral group of $D_6$

I'm having real troubles with finding the character table of the dihedral group $D_6$ of order 12: $D_6 = \langle a,b |a^6 = 1 , b^2 = 1, aba = b \rangle$. I've already found the conjugacy classes: ...
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1answer
38 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 ...
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0answers
35 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 ...
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3answers
37 views

let $G$ be dihedral group so that $D_{2n}= \langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. Prove that $N\unlhd G$ and $G/N \cong W = \{1,-1\}$

let $G$ be dihedral group so that $G = D_{2n}=\langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. 1) let $N < G$ so that $N =\langle y \rangle = \{e,y,...y^{n-1}\}$. Prove that $G \unlhd N$. 2) ...
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2answers
27 views

Finding homomorphisms

Let $G$ be the dihedral group of order 14. In the following, justify your answer. 1. Let $A=C_2$ be a cyclic group of order 2. Find all homomorphisms $G→A$. 2. Let $B=C_7$ be a cyclic group of ...
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1answer
26 views

$|G|=2p$, $p \geq 3$ prime then $G$ is abelian or $G \cong D_{2p}$

I am doing the following problem: Let $G$ be a group such that $|G|=2p$ with $p \geq 3$ prime, then $G$ is abelian or $G \cong D_{2p}$. Suppose $G$ is not abelian. By Cauchy theorem, there exist ...
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1answer
30 views

Commutator of $D_{2n}$

I am trying to calculate the commutator of the Dihedral group. If $n=1,2$ then $[D_n,D_n]=1$. Now I consider the case $n\geq 3$. I thought of using the property $[G,G] \subset H$ iff $H \lhd G, G/H$ ...
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0answers
79 views

Dihedral Group D14 - Conjugancy and Subgroups

Consider $D_{14}$, the dihedral group of order $14$. This is the group of symmetries of the regular 7-gon. Label the vertices of the pentagon clockwise as $1, 2, 3, 4, 5, 6, 7$. Let $x = (1 2 3 4 5 6 ...
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1answer
64 views

How many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads?

I am trying to figure out how many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads. I understand how to do the problem with two colors, but I am struggling to ...
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0answers
79 views

In how many different ways can one color the vertices of a regular pentagon into four colors?

I am trying to find the number of ways to color a pentagon with 4 colors up to symmetries. I know that I should be using Burnside's Theorem, and so far I know that the group $D_5$ should act on the ...
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0answers
40 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 $D_{10}$ ...
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10 views

Simplest way to see that the affine isometries of a regulara $n$-gon are linear?

What is the simplest way to see that the set of affine isometries of the plane that fix a regular $n$-gon centered at the origin are in fact linear? One can see this by showing that the origin is the ...
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0answers
16 views

Number of Sylow 2-subgroups of $D_{2n}$

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: Let $n=2^ak$ where $k$ is odd. Prove that the number of Sylow 2-subgroups of $D_{2n}$ is $k$. [Prove that if ...
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1answer
42 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 ...
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46 views

Is every symmetric group generated by some cycles?

Actually, I have two questions here: How to show that $D_3$, the dihedral group, is isomorphic to $S_3$ in details? Is every symmetric group generated by some cycles? For question (1), I know ...
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1answer
25 views

The Dihedral group $D_1$ is non-abelian?

Same as above. I'm trying to show that for any n being odd, $D_n$ has exactly n elements of order 2 where $D_n$ is non-abelian. I know that for $n\ge3$ this is true, but what about for $n=1$.
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1answer
109 views

Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
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3answers
570 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 ...
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1answer
30 views

Alternative way of proving the subgroup of rotations is normal in $\mathbb D_4$

I've just solved a basic group theory exercise which is: decide if $\{1,r,r^2,r^3\}$ is a normal subgroup of $\mathbb D_4$ (I mean the dihedral group of $8$ elements, not the one of $4$). I've used ...
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36 views

Why does $\rho_{\mathbf{Z}^t\otimes P}=\rho_R$ imply the isomorphsim of $\mathbf{Z}^t\otimes P\cong R$?

so I have happened upon a thesis regarding the calculations of various $\mathbf{Z}D_6$ modules and their isomorphisms and came across a technique which is bothering me. Let me give an example. Let ...
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25 views

Looking for Proof of the Order of a quandle cohomology group for an n-fold dihedral

Does anyone know where the proof that states that the order of a quandle cohomology group for an n-fold dihedral quandle is n is located? I have been told it can be found in one of ...
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1answer
194 views

Dihedral groups: D5

I am currently looking into structure of dihedral groups; I am interested in subgroup structure. Dihedral group have two kinds of elements; I will use their geometric meaning and call them rotation ...
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1answer
101 views

Why is the order of the subgroup 3?

I want to find the order of the subgroup $\langle ab\rangle$ of $D_3=\langle a,b\mid a^3=1,b^2=1,ba=a^2b\rangle$ According to my notes, the order of this subgroup is 3. But why is it like that? I ...
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Be $G$ a group of order $|G|=2p$ where $p$ is a prime number odd, prove that either $ G $ is cyclic or dihedral $G\simeq D_p$ the group of order 2$p$. [duplicate]

Be $G$ a group of order $|G|=2p$ where $p$ is a prime number odd, prove that either $ G $ is cyclic or dihedral $G\simeq D_p$ the group of order 2$p$. I thought the question a little difficult ...
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2answers
113 views

Given a Cayley table, is there an algorithm to determine if it is a dihedral group?

Showing that it is a group is simple enough, but is it possible to determine if it is a dihedral group or not just by looking at the Cayley table?
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1answer
131 views

subgroups of the group of pentagon symmetries

The pentagon has 5 line symmetries and therefore we will have 10 symmetries. So, we let the group G with order 10 denote the symmetry group of a pentagon. A subset $H$ of $G$ is a subgroup $(H, *)$ ...
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58 views

Understanding the irreducible representations of $D_3$

By the dimensionality theorem, $$\sum_i d_i^2 = |G|,$$ where $d_i$ is the dimension of the $i$th irreducible representation, we can infer that the dihedral group $D_3$ has two one dimensional irreps ...
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24 views

The number of subgroups of a dihedral group of an $n$-gon where $n$ is prime is equal to $n + 3$

The number of subgroups of a prime numbered dihedral group $D_{2n}$ is equal to $n + 3$. Proof: It is established that the number of subgroups of a dihedral group $D_{2n} $ (i.e the symmetries of a ...
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1answer
25 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 ...
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2answers
64 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 = ...
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1answer
23 views

What is the formula for products in dihedral groups?

Let $G \colon = \langle x, y \ | \ x^2 = y^n = e, \ x^iy^j = x^{i^{\prime}} y^{j^{\prime}} \ $ if and only if $ i = i^{\prime}, j = j^{\prime}, \ xy = y^{-1}x \rangle$. That is, let $G$ be the ...
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Lines of reflection in a Dihedral group, Why is this paradox happening?

So, we know that in a Dihedral Group $D_n$, if $r$ represents counterclockwise rotation by $2\pi/n$ radians and $s$ is any axis of reflection, then the elements of reflection stand as follows: {$s, ...
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$D_6$, regular hexagon.

Find a subgroup of $D_6$ where $D_6$ is the regular hexagon, with 12 symmetries. $$ D_6 = \{ e, r, r^2, r^3, r^4, r^5, a, ar, ar^2, ar^3, ar^4, ar^5\} $$ where $r^6 = e$. And the $r^n$ represent ...
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3answers
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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 ...
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3answers
44 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 ...
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3answers
113 views

How does one enumerate $\mathrm{Aut}(G)$, or at least compute $|\mathrm{Aut}(G)|$?

I know that the automorphisms in $\mathrm{Aut}(G)$ preserve the order of elements of $G$, so if $G$ is partitioned according to $\mathrm{Ord}$ (order), the product of the cardinalities of the ...
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1answer
102 views

Isomorphism between dihedral group and a subgroup of $S_n$

I need to find an isomorphism between $D_n$ (all symmetries of an $n-gon$) and a subgroup of $S_n$. I know that Cayley's theorem gives a nice isomorphism that shows that $D_n$ is isomorphic to a ...
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0answers
37 views

Finding conjugacy classes of $D_{10}$

Looking at the group $D_{10}$, I have found that for some (non-identity) rotation $\rho$ its centraliser has order 5, and for some reflection $\tau$ its centraliser has order 2. By the ...
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1answer
69 views

Extension of dihedral group to higher dimensions

The dihedral group $D_{2n} = \{x, y \mid x^2=y^n=yxyx=1\}$ is tied with the symmetries of the regular polygon on a plane. What is the natural extension to higher dimension? For instance, in $3$D, does ...
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3answers
109 views

Is this identity for the Dihedral group correct?

Let $D_n$ represents the Dihedral group with $2n$ elements, and my question(based on some physics backgrounds) is: Does $Z_2$ a normal subgroup of $Q_8$? If it is, then is the indentity $D_2\cong ...
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2answers
70 views

Prove that the lattice graph of $D_{16}$ is not planar

How do we prove that the lattice graph of $D_{16}$ is non-planar? I wanted to prove it using Kuratwoski's Theorem but was unable to do it. And to add one more question, are there any interesting ...
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0answers
224 views

Let G be the dihedral group of order 14

Let G be the dihedral group of order 14 i) Let A=$C_2$ be a cyclic group of order 2. Find all homomorphisms $G\to A$ ii) Let B=$C_7$ be a cyclic group of order 7. Find all homomorphisms $G\to B$ I ...
2
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1answer
50 views

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

The center of $D_6$ is isomorphic to $\mathbb{Z}_2$. I have that $$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$ $$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$ ...
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0answers
227 views

Symmetries of a regular tetrahedron

Let $G$ be the group of symmetries of a regular tetrahedron $T$, including orientation-reversing symmetries. (a) Decompose the set of faces of $T$ into orbits, and describe the stabiliser of a face. ...
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0answers
75 views

Problems related to symmetry

I am currently studying the chapter entitled "Symmetry" from Michael Artin's book "Algebra" and am having some difficulties understanding the material. It is dealing with isometries, dihedral groups, ...