# Questions tagged [dihedral-groups]

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

384 questions
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### Finding basis for a representation of $D_8$.

Let $G=D_8=\langle a,b\mid a^4=b^2=1,b^{-1}ab=a^{-1}\rangle$. The character table of $D_8$ is known and is Let $U:=\bigg\{\sum\limits_{1\leq i<j\leq 4} a_{ij}x_ix_j\mid a_{ij}\in\mathbb{C}\bigg\}$ ...
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### The number of groups $G$ (up to isomorphism) such that $G/\mathbb{Z}_3\cong D_{2n}$

I am trying to find the number of groups $G$ (up to isomorphism) such that $G/\mathbb{Z}_3\cong D_{2n}$, where $\mathbb{Z}_3$ denotes the cyclic group of order $3$ and $D_{2n}$ denotes the dihedral ...
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### 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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### Find a formula for number of orbits under action of $D_{4}$

We colour each side of a square with $k \geq 1$ colours. Find a formula for the number of orbits under the action of $D_{4}=\{ e , r,r^{2},r^{3},s,sr,sr^{2},sr^{3} \}$ on the set of colours. Now as ...
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### Conrad's $\mathit{Dihedral\ groups}$: Rigid motions taking a regular $n$-gon back to itself carry vertices to vertices

I have been reading Keith Conrad's expository paper Dihedral groups I and I have two questions about Theorem $2.2$, which deals with the size of $D_n$. In the first part of the proof you can read ...
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### Set of Rotations Cyclic?

For the dihedral group $D_{n}$ of order $2n$, is the group $R$ formed by its $n$ rotations cyclic in general? Or is the factor group $D_{n}/R$ cyclic? I am trying to show the series $D_{n}>R>(1)$...
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### Dihedral groups non-commutavity

Here is a result about dihedral groups. $rs = sr ^{-1}$, where $r$ is a rotation of $\frac{2 \pi}{n}$ radians and $s$ is a reflection about the line of symmetry from vertex $i$ and the origin. This is ...
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### How can I show that $D_{2n} \cong C_n \rtimes C_2$

Let $D_8 := \langle a,b \mid a^4 = 1 = b^2, bab = a^{-1}\rangle$ I'm trying to formally show that $$D_{8} \cong C_4 \rtimes C_2 = \langle s\rangle \rtimes \langle t \rangle$$ My book gives as hint ...
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### Show that $D_{33}$ is not isomorphic to $D_{11} \oplus Z_{3}$.

Goal: Show that $D_{33}$ is not isomorphic to $D_{11} \oplus Z_{3}$. They are both non-cyclic groups of order $66$. The same orders are possible for their elements. Comparing massive Cayley Tables is ...
I'm having problems understanding the excersice: E) Define $D_n$ as the group of symmetries of a regular n-gon. Name the vertices $V=\{V_0,V_1,...,V_{n-1}\}$ so that $$V_{k}=\exp({i\cdot\dfrac{2\pi k}{... 2answers 48 views ### Confusion about a proof about subgroups of dihedral groups This article shows that every subgroup of D_n = \langle r, s \rangle is cyclic or dihedral. Theorem 3.1. Every subgroup of D_n = \langle r, s \rangle is cyclic or dihedral. A complete listing ... 1answer 45 views ### Certain Isomorphic Representations of the dihedral group D_{3} Using the following presentation of the dihedral group D_{3} $$D_{3} = \left\langle r,s \mid r^{2} = s^{2} = (rs)^{3} = e \right\rangle$$ There is one (... 1answer 102 views ### Non-orthogonal matrix representation of dihedral groups I found this table on here: a is the first element of a dihedral group D_n, i.e. the rotation by an angle 2\pi/n. Few questions on that: 1) Non real, non orthogonal representation matrices. ... 2answers 242 views ### Group actions of D_5 I have to give 5 examples of D_5 acting on a set. So far, I have D_5 acting on the set of vertices of a pentagon and “rotating” each vertex one to the right, sending the vertices to a reflection ... 0answers 152 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{... 1answer 58 views ### Find representative for each conjugacy class of D_{10} How do you find representative for each conjugacy class of D_{10} = \langle r \rangle_5 \rtimes \langle s \rangle_2? I know D_{10} has 4 conjugacy classes which are: [Id], [r], [r^2], ... 2answers 26 views ### Nonabelian dihedral groups and a question in number theory [duplicate] I'll use a concrete definition of a dihedral group D_{2n} which emphasizes its group structure: D_{2n} consists of distinct elements r_0,...,r_{n-1},s_0,...,s_{n-1} so that for any i \in \... 1answer 38 views ### Compute a set S given information about how it is acted upon transitively by D_8 Let D_8=D_{2 \cdot 4} be the dihedral group on a regular 4-gon. Suppose that S is a subset of S_4, such that S contains the element ( 1 \ 2 \ 3). We also know that D_8 acts transitively ... 1answer 46 views ### Let D_{2n} be the dihedral group of order 2n. Let H be the set of rotations of the regular n-gon. Is H\lhd D_{2n}? [closed] Let D_{2n} be the dihedral group of order 2n, i.e., the group of symmetries of the regular n-gon. Let H be the set of rotations of the regular n-gon. Is H\lhd D_{2n}? 1answer 84 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 ... 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 ... 1answer 47 views ### Unique factorization of dihedral group My goal is to prove the following about the dihedral group D_{2n}: Prove that every element in D_{2n} has a unique factorization of the form a^{i}b^{j}, where 0 \leq i < n and j=0 or ... 1answer 38 views ### A formula for the number of order 2 elements of D_m\times D_n for even m>2 and odd n>2. (Gallian 8.24.) This is Exercise 8.24 of Gallian's "Contemporary Abstract Algebra (Eighth Edition)". Answers that use material from the textbook prior to the exercise are preferred. Presentations, for instance, are ... 1answer 35 views ### Group Theory - Dihedral Groups Two questions related to Dihedral groups: What is the conventional notation for Dihedral groups? Is it Dn where n is the number of sides in a regular n-gon, or is it D2n where n is the number of ... 1answer 67 views ### Does D_4 have a verbal subgroup of order 4? Does D_4 have a verbal subgroup of order 4? How did this question arise: In the comments Q_8 ad D_4 were pointed to be a possible counterexample to this question: Is it true, that for any two ... 0answers 21 views ### Showing two series are not the same. I want to show that the following two composition series are not the same: D_{8}\triangleright \left \langle s,r^{2} \right \rangle \triangleright \left \langle s \right \rangle \triangleright (1) ... 2answers 86 views ### Find a group G with a\in G such that |a|=6 but C_G(a)\neq C_G(a^3). This is part of Exercise 46 of Chapter 3 of Gallian's "Contemporary Abstract Algebra". Notation 1: The centraliser of g in a group G is denoted C_G(g). Notation 2: The dihedral group ... 0answers 14 views ### Coloring triangular dihedral #2 To start with, my dihedral is a bit specific, here is a picture I need to find amount of ways to color faces ( there are 8 ) into 3 colours. I have already something in my mind because of help ... 2answers 81 views ### Quotient group of the dihedral group by \langle r^2 \rangle. Show that G/H is abelian, where G is the dihedral group$$ G={\langle r,\, f \mid r^n=f^2=1,\, rf=fr^{-1}\rangle} and $H$ is the subgroup $\langle r^2 \rangle.$ I've tried showing that for $... 0answers 21 views ### Derived series of the Dihedral group [duplicate] I'm working on derived subgroups because I'm studying for an exam and I want to show that in the case of the dihedral group$D_{2n}=\langle\sigma ,\tau|\sigma^n=\tau^2,\sigma^{\tau}=\sigma^{-1}\rangle$... 1answer 52 views ### Coloring sides of truncated triangular dihedral(bipiramid) into 3 colours I need to find out the amount of ways to colour truncated triangular dihedron into 3 colours. So, the task will be easier if I had simple triangular dihedron. First of all, do I understand right ... 2answers 83 views ### Find$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$G/Z(G)$... 0answers 66 views ### If$G$has a nontrivial centre, must every subgroup of index$3$be normal? If a group$G$has a nontrivial centre, must every subgroup of index$3$be normal?$S_3$yields an example of a group with a non-normal subgroup of index$3$, although it has a trivial centre. ... 1answer 48 views ### Automorphism of$D_8$[duplicate] I am trying to prove that$Aut(D_8) \equiv D_8$. It is not hard to see that$\lvert Aut(D_8)\rvert = 8$. Indeed, it is at most$8$as$r$(canonical rotation) has order$4$and$s$(canonical ... 2answers 36 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 ...
If a square remains fixed in the plane, how many different ways can the corners of the square be colored if three colors are used? Why does the answer use $D_{4}$ when the square cannot move? I don't ...