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

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I need to prove that there is one homomorphism $\varphi : Dn \to Sn$ such that

I need to prove that there is one homomorphism $\varphi : Dn \to Sn$ such that $\varphi$($\tau$) = $$ \begin{pmatrix} 1 & 2 & . & .& . & n \\ 2 & 3 ...
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2answers
64 views

Is the infinite dihedral group an inverse limit of the finite dihedral groups?

The p-adic numbers are the inverse limit of the rings $\mathbb Z / p^n \mathbb Z$. Can the infinite dihedral groups be construed as some sort of inverse limit of finite dihedral groups?
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Elements of Dihedral Groups

Is there an elegant way of showing that the elements of a dihedral group are only rotations and reflections? Specifically, I'm having trouble convincing myself that a composition of a rotation and a ...
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1answer
27 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 ...
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1answer
28 views

Finding subgroups of $D_{2p}$

Let $p$ be an odd prime Find all the subgroups of $D_{2p}$. We know that all $g^i$ $(i=1,\dots,p-1)$ have order $p$ and all $g^ih$ $(i=0,\dots,p-1)$ has order $2$. By Lagrange if $H < G$ then ...
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57 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 ...
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28 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 ...
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38 views

Subgroups of the dihedral group D_n modulo Aut(D_n)

This question is related to this math.se question. Consider the dihedral group $D_n = \langle r,s \rangle.$ Two subgroups $G, H \leq D_n$ are said to be ''isomorphic'' if there is an $f \in ...
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1answer
47 views

Group generated by self-inverse elements

Given objects $x_1, \dotsc, x_n$, is there a name for the group generated by $x_1,\dotsc,x_n$ subject only to the relations $x_i^2 = 1$ for all $i \in \{1,\dotsc,n\}$? The dihedral group seems ...
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1answer
43 views

Proving that the dihedral group $D_n$ has $2n$ elements

I am trying to prove that the dihedral group $D_n$ has $2n$ elements by using the theory of group actions. Specifically I want to use the orbit stabilizer theorem. So I need $D_n$ to act on a specific ...
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$Aut(D_4)$ is isomorphic to $D_4$?

Problem statement: I need to find out if $Aut(D_4)$ is isomorphic to itself($D_4$) 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 ...
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1answer
48 views

Simple homomorphism Question: Prove that $φ(s)$ is a reflection

Let r ∈ D20 be an element of order 20 and let s ∈ D20 be a reflection. Suppose that φ : D20 → D20 is a homomorphism such that φ(r) = r^12 Prove that $φ(s)$ is a reflection
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2answers
60 views

Number of normal subgroups from $F_{2}$ which factor groups are isomorphic to $D_{n}$

What is the number of normal subgroups of the free group $F_{2}$ whose factor groups are isomorphic to the dihedral group $D_{n}$?
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1answer
40 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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1answer
18 views

Motions that Leave a Prism Invariant

Identify the group of all motions that leave a right prism invariant. It seems like the normal dihedral group, $D_n$, with respect to the base would be a subgroup of this group. That is to say that ...
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29 views

History of Dihedral Groups?

I'm trying to write a brief historical outline for a project concerning Dihedral Groups, but It has been to hard for me to find information about the first time these groups appeared on literature. If ...
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1answer
37 views

Is there any easier way to find out every proper subgroups of Dihedral group 4?

There are 8 elements in Dihedral group 4 When finding proper subgroups of this, do I have to list every element and draw a table to find out subgroups ?
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1answer
56 views

How many proper nontrivial subgroups do D5 have?

Do I have to find out every element in D5 and draw a table to find out subgroups? I know how to find out every single element in D5, but can't think of how to find proper nontrivial subgroups
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1answer
44 views

Isomorphic relation between dihedral groups

Theorem Let $G,H$ be abelian groups such that $Dih(G)\cong Dih(H)$ If $G$ is finitely generated, then $G\cong H$. I'm curious whether "finitely generated" hypothesis can be removed. If ...
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Definition of Dihedral group via semidirect product

Let $G$ be an abelian group and let $\varphi:Z_2\rightarrow Aut(G)$ be the homomorphism where $\varphi(\overline{1})$ is the inversion automorphism. Define $Dih(G)=G\rtimes_{\varphi} Z_2$. Now set ...
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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
37 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
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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
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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
124 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
53 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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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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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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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
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$|G|=2p$, $p \geq 3$ 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
36 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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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
147 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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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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|${G_x}$| 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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1answer
52 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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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
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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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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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745 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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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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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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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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322 views

Subgroup structure of dihedral groups

I am currently looking into structure of dihedral groups; I am interested in their subgroup structure. Dihedral group have two kinds of elements; I will use their geometric meaning and call them ...
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1answer
107 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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200 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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222 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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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 ...