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

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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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0answers
34 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, ...
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50 views

$D_6$, regular hexagon.

Find a subgroup of $D_6$ where $D_6$ is the regular hexagon, with 12 symmetries. $$ D_6 = \{ e, r, r^2, r^3, r^4, r^5, a, ar, ar^2, ar^3, ar^4, ar^5\} $$ where $r^6 = e$. And the $r^n$ represent ...
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3answers
35 views

Intuition - $fr = r^{-1}f$ for Dihedral Groups - Carter p. 75

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square's vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles's $f$ is different. Carter fleshes out why ...
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3answers
36 views

Showing an Isomorphism between question group of $S_4$ and $D_6$

I have a subgroup $N$ of $S_4$, where $ N = [1, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)] $ I need to explain whether quotient group $G/N$ is isomoprhic to either $C_6$ or $D_6$ (no proof required, just an ...
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3answers
88 views

How does one enumerate $\mathrm{Aut}(G)$, or at least compute $|\mathrm{Aut}(G)|$?

I know that the automorphisms in $\mathrm{Aut}(G)$ preserve the order of elements of $G$, so if $G$ is partitioned according to $\mathrm{Ord}$ (order), the product of the cardinalities of the ...
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1answer
63 views

Isomorphism between dihedral group and a subgroup of $S_n$

I need to find an isomorphism between $D_n$ (all symmetries of an $n-gon$) and a subgroup of $S_n$. I know that Cayley's theorem gives a nice isomorphism that shows that $D_n$ is isomorphic to a ...
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0answers
29 views

Finding conjugacy classes of $D_{10}$

Looking at the group $D_{10}$, I have found that for some (non-identity) rotation $\rho$ its centraliser has order 5, and for some reflection $\tau$ its centraliser has order 2. By the ...
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1answer
57 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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3answers
99 views

Is this identity for the Dihedral group correct?

Let $D_n$ represents the Dihedral group with $2n$ elements, and my question(based on some physics backgrounds) is: Does $Z_2$ a normal subgroup of $Q_8$? If it is, then is the indentity $D_2\cong ...
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38 views

The dihedral group $D_n$ of order $2n$, $n\geq 3$, has a subgroup of $n$ rotations and a subgroup of order $2$. [closed]

The dihedral group $D_n$ of order $2_n$ ($n\geq 3$) has a subgroup of $n$ rotations and a subgroup of order $2$. Explain why $D_n$ cannot be isomorphic to the external direct product of two such ...
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2answers
44 views

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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0answers
146 views

Let G be the dihedral group of order 14

Let G be the dihedral group of order 14 i) Let A=$C_2$ be a cyclic group of order 2. Find all homomorphisms $G\to A$ ii) Let B=$C_7$ be a cyclic group of order 7. Find all homomorphisms $G\to B$ I ...
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1answer
40 views

Center of $D_6$ is $\mathbb{Z}_2$

The center of $D_6$ is isomorphic to $\mathbb{Z}_2$. I have that $$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$ $$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$ ...
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2answers
95 views

True or wrong: Two statements on finite groups [closed]

Determine, with explanation, if the following are always true. a) If $P$ and $Q$ are each $p$-Sylow subgroups of a group $G$, then either $P=Q$ or $P\cap Q=\{e\}$. b) If $G$ is a group ...
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0answers
167 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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64 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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1answer
114 views

Cosets and finite groups of orthogonal operators on the plane

Can someone show me how to compute the left cosets of the subgroup $H=\{1, x^5\}$ in the Dihedral group $D_{10}$ I know that $D_{10}$ is generated by two elements $x$ and $y$ such that: $$x^{10}=1, ...
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95 views

Could someone explain chirality from a group theory point of view?

While answering this question my interest in the rotation/reflection group was piqued. I personally know very basic group theory, not much more than what a group really is. I understand that the ...
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1answer
93 views

Subgroups of $D_4$

I need to determine the subgroups of the dihedral group of order 4, $D_4$. I know that the elements of $D_4$ are $\{1,r,r^2,r^3, s,rs,r^2s,r^3s\}$ But I don't understand how to get the subgroups..
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1answer
83 views

Expressing a product in a Dihedral group

Write the product $x^2yx^{-1}y^{-1}x^3y^3$ in the form $x^iy^j$ in the dihedral group $D_n$. I used the fact that the dihedral group is generated by two elements $x$ and $y$ such that: $y^n=1$, ...
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0answers
148 views

How many elements of order 2 are in the Group D10 x D8?

Dihedral groups really confuse me. What are the elements of D10 and D8? I need to know these in order to find the elements in D10 x D8. Then maybe I can figure out the order of those elements. Any ...
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1answer
57 views

An element of order $n$ generates a normal subgroup of $D_n$

Let $a$ be an element of order $n$ of $D_n$. Show that $\langle a\rangle \lhd D_n$ and $D_n/\langle a\rangle \cong \mathbb Z_2$. Proof: Let $K = <a>$ for some a ∈ G. Let H ≤ K be an ...
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1answer
23 views

Why isn't $r^{\frac{n}{2}}$ classified as an involution?

Suppose you have a $(2n)$-gon for some $n \in \mathbb{N} : n > 1$. Then the rotation $r^{\frac{n}{2}}$ where $n$ is the number of vertices imposed on the $(2n)$-gon is the same as an involution. ...
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1answer
34 views

Question about definition of generator for dihedral groups?

Dummit and Foote mention that a relation for the dihedral group is $rs = sr^{-1}$. Now, I have interpreted the statement to mean $r$ is a rotation of $\frac{2\pi}{n}$ radians and $s$ is an ...
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3answers
77 views

Is this Cayley table correctly computed? If so, is it correct way of presentation of this dihedral group?

Is the following table for $D_4$ correct? $$\begin{array}{c|c|c|c|c|c|c|c|c|} * & 1 & g & g^2& g^3& f& fg & fg^2 & fg^3 \\ \hline 1 & 1 & g & g^2 ...
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1answer
44 views

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; \ ...
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6answers
564 views

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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1answer
48 views

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 ...
0
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1answer
105 views

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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1answer
22 views

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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1answer
53 views

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 ...
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1answer
91 views

Prove: $D_{8n} \not\cong D_{4n} \times Z_2$.

Prove $D_{8n} \not\cong D_{4n} \times Z_2$. My trial: I tried to show that $D_{16}$ is not isomorphic to $D_8 \times Z_2$ by making a contradiction as follows: Suppose $D_{4n}$ is isomorphic ...
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3answers
148 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 ...
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1answer
60 views

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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1answer
125 views

Dihedral group as a matrix group

I wish to consider the dihedral group as a matrix group. One way to do that is to consider it as a finite subgroup of $O_2$, a group of orthogonal $2\times 2$ matrices, defined by ...
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2answers
93 views

Number of conjugacy classes of the reflection in $D_n$.

Consider the conjugation action of $D_n$ on $D_n$. Prove that the number of conjugacy classes of the reflections are $\begin{cases} 1 &\text{ if } n=\text{odd} \\ 2 &\text{ if } ...
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1answer
92 views

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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0answers
53 views

Induced character for D12

Let $H$ is subgroup of $G=D_{12}$ was generated by $R^3$ and $S$ ($R$ is rotation and $S$ is a reflection). We know that This subgroup is isomorphic to the dihedral group $D_4$. Suppose $F$ is the ...
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3answers
214 views

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 ...
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1answer
268 views

Composition Series for Dihedral Groups

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 ...
3
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2answers
488 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 ...
3
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1answer
290 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, ...
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2answers
171 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( ...
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3answers
423 views

On the centres of the dihedral groups

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 ...