Questions tagged [dihedral-groups]

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

385 questions
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Two questions about the dihedral group

First question: 1) Is the sum of subgroup indices of dihedral group with $2n$ elements equal to $\sigma_2(n)+2\cdot \sigma(n)$? Second question: 2) Is $\sigma_2(n)+2\cdot \sigma(n) \le L(H(D_n))$? ...
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classification of representations of $D_{1009}$

A follow-up of this question To fix ideas, take $n=1009$. $D_n$ has $2$ irreducible representations of degree $1$ and $504$ representations of degree $2$. Are the degree 1 representations all ...
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Orders of the Elements of $D_6/Z(D_6)$

I have been trying to calculate the orders of the elements of $D_6/Z(D_6)$. For example, using $R_{60}$ to represent rotation by 60 degrees and $R_0$ to represent rotation by 0 degrees (the identity ...
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Question about conjugacy classes in dihedral groups [duplicate]

I'm trying to find the conjugacy class of a rotation $r^{k}$. Is it unitary? How about a symmetry $s$? Any ideas?
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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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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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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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Understand the free group universal property applied to $D_n$

For $n ≥ 3$ and $D_n$ the dihedral group of order $2n$ with présentation $\langle r, s : r^n = s^2 = srsr = 1\rangle$ prove that for all $(a, b) \in (\Bbb Z/n\Bbb Z)^2$, there exists a morphism $f$ ...
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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 ...
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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 ...
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 ...