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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elements of order two in $D_{10}$

Which elements have order two in $D_{10}$? In $D_{10}$ there are $10$ elements, five of which are rotations and five reflections. Let $\rho = (1\hspace{1mm}2 \ldots 5)$ and $\tau = (1)(2\hspace{1mm}5)...
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1answer
30 views

Conjugate of elements in the Dihedral Group

I was trying to do the following from a past exam of my Rings and Groups' professor Classify all conjugacy classes of the elements in the dihedral group $D_n$ = $\{ 1,r,r^2, ... , r^{n-1} ,s ,rs ,...
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1answer
85 views

How to show that $|D_{2n}| = 2n$ via the presentation?

Consider the dihedral group $$D_{2n}= \langle a,b \mid a^n = 1 = b^2, b^{-1}ab = a^{-1}\rangle$$ How can I show that $|D_{2n}| = 2n$? I'm trying to show that we can write every element in the form ...
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1answer
37 views

$G_{x_1x_2+x_3x_4}\cong D_8$

Let $R=F[x_1,x_2,x_3,x_4]$ be the set of polynoms in 4 variables over a field $F$. Let a map $\varphi:S_4\to \operatorname{Sym}(R)$ by $$ \\ (f(x_1,x_2,x_3,x_4))\varphi(\sigma)=f(x_{1\sigma},x_{2\...
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1answer
49 views

Orbit of conjugation on subgroups of $D_8$

Let $X$ be the set of all subgroups of $D_8$ with order $2$. For fixed $g\in D_8$, and for all $x\in X$, conjugation by $g$ is defined by $$x\mapsto gxg^{-1}$$ What is the orbit of this group action? ...
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1answer
171 views

Group of Rotations Stabilizer in D4

In D4, the subgroup of rotations is not a stabilizer for any point in a square (even the center). Am I missing anything? Thanks! Edit: To clarify, I wanted to ask if the subgroup of rotations of D4 ...
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2answers
139 views

Quotient Group $D_2n/R$ is abelian where $R$ is the group of rotations?

$D_2n$ is not abelian. However, the group of rotations, denoted $R$, is. I've already shown that $R$ is a normal subgroup of $D_2n$; however I'm stuck at showing the quotient group is abelian. I know ...
2
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2answers
82 views

How can we find all the subgroups?

I want to find all the normal subgroups of $D_n$. We have that $K$ is a normal subgroup of $D_n$ iff $$gkg^{-1}=k\in K, \forall g\in D_n \text{ and } \forall k\in K$$ right? Could you give me some ...
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1answer
2k views

Dihedral groups are solvable [duplicate]

I'm trying to prove the dihedral groups are solvable for any Dn. I use the normal subgroup of all rotations, since the quotient of Dn/{rotations} is isomorphic to Z2 so it's abelian as well, so we get ...
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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? ...
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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 ...
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1answer
222 views

Show that there exist an injective homomorphism of dihedral group $D_n$ into $G$.

Let: $$n\gt2 \; \text{and the group} \; (G,⋅)$$ Consider that there existe: $a,b∈G$ such that $a^n=b^2=1_G$ and $b⋅a=a^{−1}⋅b$ and $n$ is the smallest $n≥1$, such that $a^n=1_G$. ...
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1answer
86 views

Counting elements of order 6 in $D_{12} \times Z_2$

Actually I know how to count number of elements of particular order from direct product .But In this my counting is not matches with answer so I wanted to know where is my mistake lies . I wanted to ...
2
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1answer
151 views

Is there a non-trivial homomorphism from $D_4$ to $D_3$?

The question is actually a bit broader than this, but it is a good starting point. Given two Dihedral groups $D_4$ and $ D_3$ we wish to construct a nontrivial homomorphism $f:{ D_4\to\ D_3}$. Is ...
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2answers
78 views

Presentation of $D_4$

When one writes $D_4=\langle r,s\mid r^4=s^2=1,rsrs=1\rangle$ they are describing a quotient group. Let $S=\{s,r\}$ and $R=\{r^4,s^2,rsrs\}$. $$F_S=\langle r,s\rangle,\quad R^{F_S}=\{grg^{-1}\mid r\in ...
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1answer
49 views

Need help with a proof involving subgroups of $D_{n}$

I have a question that needs attention regarding the dihedral group $D_{n}$. Here is the context of the problem: Consider the Dihedral group $D_{n}$. Let $\sigma$ be a rotation counterclockwise by $\...
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1answer
50 views

Presentation of $D_4 \rtimes Aut(D_4)$

Currently, I am reading Representation Theory of Semi-Direct Products by Reyes. In section $6$, the author mentions that the presentation of $D_4 \rtimes Aut(D_4)$ is as follows. $$ D_4 \rtimes Aut(...
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1answer
50 views

Construct groups that satisfy given conditions

I want to construct groups which satisfy the following conditions. It should be generated by an abelian normal subgroup $M$ and a subgroup $G$. $G$ should be either an Abelian group or Symmetric ...
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1answer
298 views

Conjugacy classes, irreducible representations, character table of $D_{10}$ (order 20)

$D_{10}=\langle r,s \rangle$ is the dihedral group of order 20 . I have been struggling a bit with this question, particularly c, regarding values in the character table (a). Find the conjugacy ...
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1answer
65 views

Is there a surjective homomorphism from $D_{10}$ to $\mathbb Z _2\times \mathbb Z_2$

Is there a homomorphism $f\colon D_{10} \to \mathbb Z_2\times \mathbb Z_2$ that is onto? Attempt: $D_{10} = \{e,s,r,...,r^9,sr,...,sr^9\}$, $\mathbb Z_2\times \mathbb Z_2 = \{(0,0),(1,0),(0,1),(1,1)\...
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2answers
142 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
66 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 it'...
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1answer
27 views

Non-trivial isomorphism between the dihedral group to itself.

I want to find a non-trivial isomorphism between the dihedral group $D_n$ and itself. Non-trivial means that the isomorphism won't be the identity. I looked at the group $D_n$ as the set of the ...
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1answer
101 views

No common eigenvectors then representation irreducible

Embed an equilateral triangle into $\mathbb{R}^2$ with vertices $(1,0), (\frac{-1}{2}, \frac{\sqrt{3}}{2}), (\frac{-1}{2}, -\frac{\sqrt{3}}{2})$. Counterclockwise rotation and reflection over the $x$ ...
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0answers
41 views

Minimal amount of information required to define a Dihedral group?

Given the dihedral group $D_n$, where $r$ is a single rotation, and $s$ is a reflection, I must show that $s \circ r \circ s = r^{-1}$. $D_n$ is a group, with properties: 1) $1 = r^0 = s^0$ 2) for ...
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0answers
74 views

Find a topological space whose fundamental group is $D_4$ [duplicate]

$D_4$ here indicates the dihedral group of order 8. Does there exist any trivial example of topological spaces such that it has $D_4$ as it's fundamental group.
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112 views

Action of $D_{2n}$ on pairs of opposite vertices of a regular n-gon

Having some difficulty understanding the solution to this problem. We have an action of $D_{2n}$ on set $A=\left\{{\left\{\overline{a},\overline{k+a}\right\}\mid1\leq a\leq k}\right\}$ and $s.\left\...
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1answer
231 views

Questions about group representation of infinite dihedral group

I am learning group representation theory, and I have troubles understanding the applications of group representation. Could you please help me to give explain the following question? My question is:$...
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1answer
80 views

What are dihedral groups?

I have trouble understanding what exactly a dihedral group is. I read about how they are rotations and reflections along the faces of a polygon. But then what does that mean? Whatever you do to a ...
2
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1answer
137 views

Finding normal subgroups

Let $G=\langle e, r,..., r^{n-1},s,sr,...,sr^{n-1} \rangle $ be a dihedral group with $2n$ elements, for $ 3 \leq n$. Prove that the only normal subgroups of $G$ are $\langle r^d \rangle$ (where $d$ ...
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0answers
59 views

Dihedral groups acting on Riemann surfaces

I'm studying the quotient riemann surface $X/G$. I'm looking for examples of dihedral groups $D_n$ acting on some riemann surfaces $X$ or at least acting on it's Jacobian JX. Does anybody knows some ...
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0answers
357 views

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

Let $2n = 2^ak$, where $k$ is odd. We wish to show that the number of $2$-Sylow subgroups of $D_{2n}$ is $k$. My approach has been to construct such a $2$-Sylow subgroup $P_2$, and then show that the ...
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1answer
47 views

How shall we show that the only possible orders of any element in dihedral group $D_n$ will be either a divisor of 2 or $n$?

I am studying the dihedral group $D_n:=\{r_n, f_n: r_n^n=f_n^2=(r_nf_n)^2=e_n\}$. I am willing to show that the possible orders of any element in it will be either a divisor of 2 or $n$. But don't ...
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0answers
69 views

How can I show that $D_{2n}$ follows from these relations?

Suppose we have a group $A$ which is generated by generators $R$ and $F$, subject to the relation $$ R^n=I, F^2=I,RF= FR^{-1}.$$ It should be just the dihedral group of order $2n$, the one generated ...
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0answers
154 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 \rm{...
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1answer
171 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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0answers
795 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
100 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, ....
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3answers
2k views

Prove $rs=sr^{-1}$ in ${\rm Dih}(2n)$

Let $r$ and $s$ be the rotation and reflection symmetries respectively in ${\rm Dih}(2n)$, the dihedral group of order $2n$. Show that $rs=sr^{-1}$. I also need to show by induction that $r^js=sr^{-j}...
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5answers
302 views

Defining dihedral groups $\{\sigma \in S_n: $ something $\}$

I am trying to understand hos one can define the dihedral groups $D_n$. I have seen the "definition" that just says this is the group of symmetries of an $n$-polygon. So you have rotations and ...
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3answers
163 views

Help with understanding proof: every group of order 6 is cclic or dihedral.

I am considering the proof that every group of order 6 is cyclic or dihedral (following some lecture notes). The initial part of the proof, in broad brushstrokes, considers that the elements in the ...
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2answers
104 views

Prove a group that has a normal subgroup isomorphic to $D_8$ has a non-trivial center

Let $G$ be a group which has a normal subgroup isomorphic to $D_8$. Prove that $G$ has a non trivial center. So, given $g\in G$, $h\in D_8$ $ghg^{-1}\in D_8$. So I tried to prove that there is an ...
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2answers
203 views

Group Theory- symmetry group

The Symmetric group of the set ${1,2...,n}$ is $S_n$. It is the set of permuations of the set ${1,2...,n}$. But what is the symmetry group of a polygon? or a $3D$ shape? For example I saw: "A ...
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1answer
31 views

Why is $\langle r\rangle$ characteristic in $D_n$?

I need to determinate if $\langle r\rangle$ is characteristic in $D_n = \langle r \rangle_n \rtimes \langle s \rangle_2$. This is trivial if I use the result that every cyclic group is ...
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2answers
37 views

What is $D_{16}/ Z(D_{16})$?

I was asked the following: Let $D_{16}$ be the dihedral group of order $16$. What is $D_{16} / Z(D_{16})$? I know that the center of $D_{16}$ har order $2$. So therefore, the quotient has order $16/2 ...
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2answers
67 views

Is $G$ isomorphic to the dihedral group $D_{10}$?

Let $G\le S_6$ be the subgroup generated by the permutations $\sigma=(12356)$ and $\tau=(26)(35)$. I'm asked to determine: (a) the order of $G$ and the period of each element; (b) if $G$ is ...
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2answers
69 views

How many distinct composition series does the group $D_{12}$ have?

How many distinct composition series does the group $D_{12}$ have? I know that $D_{12} \trianglerighteq \mathbb{Z}_6 \trianglerighteq \mathbb{Z}_3 \trianglerighteq \{e\}$ is a composition series ( ...
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1answer
44 views

Dihedral group of order 10 in GAP

The dihedral group of order $10$ is given by $D_{10} = \langle a,b| a^5 = b^2 = 1, bab^{-1} = a^{-1}\rangle$. Now I need to find all the elements in GAP. But whenever I type ...
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2answers
51 views

Dihedral group generated by $\langle r,s\rangle$ for all $n$

Under wikipedia for Dihedral groups it claims the following: The $2n$ elements in $D_n$ can be written as $\{e,r,r^2,r^3,\ldots,r^{n-1},s,rs,r^2s,\ldots,r^{n-1}s\}$. I know why this is true and it ...
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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 $G/Z(G)$ ...