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

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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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9 views

Quotient of a semidihedral group

For instance, we know that quotient of a dihedral group is dihedral. Please how can we characterize the quotient of a semidihedral/quasidihedral group? Thanks.
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26 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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0answers
35 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 ...
2
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1answer
43 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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2answers
83 views

$Aut(D_4)$ 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
46 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
59 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
39 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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0answers
27 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
33 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
54 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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3answers
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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
25 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
40 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
116 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
49 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
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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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3answers
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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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2answers
34 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
27 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 ...
3
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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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0answers
138 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
125 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
129 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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|${G_x}$| 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
26 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
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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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3answers
720 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
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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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1answer
307 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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26 views

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
191 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
212 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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78 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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1answer
39 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
73 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
25 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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57 views

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, ...