# 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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### How to describe all normal subgroups of the dihedral group Dn? [duplicate]

The dihedral group consists of rotations and symmetries. But the symmetry group is a group only if n is even, thus the group of rotations is a normal subgroup of the dihedral group. So how to ...
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### Prove that the lattice graph of $D_{16}$ is not planar

How do we prove that the lattice graph of $D_{16}$ is non-planar? I wanted to prove it using Kuratwoski's Theorem but was unable to do it. And to add one more question, are there any interesting ...
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### Prove two reflections of lines through the origin generate a dihedral group.

Let $l_1$ and $l_2$ be the lines through the origin in $R^2$ that intersect in an angle π/n and let $r_i$ be the reflection about $l_i$. Prove the $r_1$ and $r_2$ generate a dihedral group $D_n$. ...
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### Center of $D_6$ is $\mathbb{Z}_2$

The center of $D_6$ is isomorphic to $\mathbb{Z}_2$. I have that $$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$ $$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$ My ...
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### 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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### 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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### How to find this formula in this dihedral group of transformations of the plane?

In the group of all the bijections of the Euclidean plane onto itself, let $f(x,y) \colon = (-x,y)$ and $g(x,y) \colon = (-y,x)$ for all points $(x,y)$ in the plane. Let G:= \{f^i g^j | i=0,1; \ g=...
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### Is it true that a dihedral group is nonabelian?

Is it true that a dihedral group is nonabelian? I'm not sure if the result is true. I checked it for some lower order and I think the result may correct. But I failed to prove/disprove the result.
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### I'm trying to understand the following part from Gallian text

I'm trying to understand the following part (Chap. Sylow Theorem, Paragraphs preceding the article Application of Sylow Theorem) from Gallian text I'm trying to understand the why. That is I need to ...
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### Dihedral group as a direct product

In a problem from a past exam I am asked "When can $D_n = \langle r,s\mid r^n = s^2 = (rs)^2 = 1\rangle$, the dihedral group of order $2n$, be expressed as a direct product $G\times H$ of two ...
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### Elements of $D_{2n}$ in terms of isometries

In course of studying Dihedral Group I'm having trouble to get what exactly the elements of $D_{2n}$ are. According to the Dummit-Foote texts For each $n∈\mathbb Z^+,n≥3$ let $D_{2n}$ be the set ...
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### Find the inner and outermorphisms of a particular dihedral group

Given that |Inn($D_8$)| = 8 and |Out($D_8$)| = 2 where Out($D_8$) = Aut($D_8$)/Inn($D_8$) and $D_8$ = {e,r,$r^2$,..,$r^7$,s,sr,...,$sr^7$} we want to find Inn($D_8$) and Out($D_8$). We know that Out(...
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### Counting 0-1 matrices up to symmetry

I'm interested in counting the number of n×n 0-1 matrices with a given number of 1s up to rotation and reflection. What is the best way to do this if n is not too small? For example, consider 4&...
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### Dihedral group of a square $D_4$

Prove that in the $D_4$ a square's symmetry group each element can be uniquely written as $r^i s^j$, $i =1,2,3, \ \ j=0,1$, where $r$ is a rotation by $\frac{\pi}{2}$ around the centre of the square, ...
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### Examples of the dihedral group $D_4$ acting on sets

Consider the group $D_4$. Give examples of $D_4$ acting on a set. Attempt: So $|D_4| = 8$. I have come up with a few, but I was wondering what some people here thought. First one we came up with ...
I have been thinking about a composition series for $D_{14}\times D_{10}$ (where $D_{2n}$ is the dihedral group with $2n$ elements). Is the following a correct composition series for $D_{10}\times D_{... 1answer 1k views ### Field extension with dihedral Galois group In an old exam of my Galois Theory class there is the following question which troubles me: Let$p \neq 2$be a prime number and$k \geq 1$an integer. Give an example of a galois extension$L/K$... 2answers 2k views ### Algebra - Infinite Dihedral Group Let$G$be the set of bijections$\mathbb{R} \to \mathbb{R}$which preserve the distance between pairs of points, and send integers to integers. Then$G$is a group under composition of functions. The ... 1answer 2k views ### Prove that the dihedral group$D_4$can not be written as a direct product of two groups I like to know why the dihedral group$D_4$can't be written as a direct product of two groups. It is a school assignment that I've been trying to solve all day and now I'm more confused then ever, ... 2answers 3k views ### How to determine what group a Galois group is isomorphic to Consider$x^{4}-2=(x+\sqrt{2})(x-\sqrt{2})(x+i\sqrt{2})(x-i\sqrt{2}) \in \mathbb{Q}[x]$. Let$K=\mathbb{Q}(\sqrt{2},i)$be the splitting field of$x^{4}-2$. Since$K$is a splitting ... 2answers 230 views ### Understanding some strange notation for$D_4$I'm now studying Fraleigh's Abstract algebra(7th). In section 8, there is a group table for$D_4$, with some strange notations that I can't compute it easily. He uses$\rho_0=\left( \begin{array}{...
In an proof that I recently read, the following 'fact' is used, where $D_{2n}$ denotes the dihedral group of order $2n$: If $n$ is even, then $D_{2n} \cong C_2 \times D_n$. The (short) given ...