# 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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### Binary sequences of $a$ zeros and $b$ ones which have $a+b$ circular shifts.

Let $P(a;b)$ be the set of all sequences generated by permutation of $a$ zeros and $b$ ones. $$|P(a;b)|=\binom{a+b}{a}$$ I want to prove that if $gcd(a,b)=1$, every sequence in $P(a;b)$ will have $a+b$...
1 vote
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### Proving a non-cyclic $G$ has a cyclic subgroup of index $2$ and $G\cong D_k.$

Suppose $G$ is a non-cyclic group generated by two elements, both of order $2.$ Prove that $G$ has a cyclic subgroup of index $[G:C]=2.$ If such a group is of finite order $2k,$ prove it is isomorphic ...
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### Listing all subgroups of $D_{10}$

My work so far: The cyclic subgroups are trivial to find. They are simply rotations generated from $\langle r^d\rangle$ where $d\mid10$. So $d=1,2,5,10$. $\langle r \rangle= \{e,r, r^2, \dots, r^9 \}$ ...
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### Symmetries of the dihedral groups $D_n$ and $D_{nh}$.

According to my lecture book (Linear Representations of Finite Groups by Serre), the dihedral group $D_n$ consists of $n$ rotations and $n$ reflections in the plane that preserve a regular polygon ...
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### Proving that $r^i \neq sr^j$ in $D_{2n}$ [duplicate]

I was trying to proof that the order of the dihedral group $D_{2n}$ is $2n$ given its presentation $$D_{2n} = \langle r,s ~\vert~ r^n = s^2 = 1 ,~ rs = sr^{-1} \rangle$$ and I further assumed that ...
1 vote
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### Importance of abelian subgroups of dihedral group in computing conjugacy classes?

Let $D_{8}$ be the dihedral group of order 8. We would like to find its conjugacy classes. Authors Dummit and Foote write: In $D_{8}$ we may also use the fact that the three subgroups of index 2 are ...
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### Unique $p$-Sylow subgroup of Dihedral group $D_n$.

If $p>2$ is a prime number that divides $n \in \mathbb N$, then $D_n = \langle r,s \rangle$ (with $r^n=s^2=id, sr=r^{-1}s$) has a unique $p$-Sylow subgroup $H \leq D_n$. I'm having trouble with ...
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### What do the dotted lines represent in the hexagonal lattice system?

The above is one of the four two-dimensional lattice systems. See this article for some info. The parallelogram is the primitive unit cell, because it can be translated in order to recreate the entire ...
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### Representation of symmetry group $D_4$ acting on a square

Consider the square with vertices $(\pm 1, \pm 1)$. The symmetry group $D_4$ of this square acts by permuting the vertices Show that each permutation comes from a linear transformation. Compute the ...
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### Let $n=dm$ and consider the group $D_n$ , with $K=⟨ρ^m⟩$ .Show that the following $2m$ cosets of $K$ are all distinct:

$D_n = \langle \mu, \rho \mid \mu^2 = \rho^n = \epsilon, \mu\rho = \rho^{-1}\mu \rangle$ Let $n=dm$ and consider the group $D_n$, with $K = \langle \rho^m \rangle$. Show that the following $2m$ cosets ...
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### Prove that all dihedral groups of degree $4$ are isomorphic

I was studying about group theory when I came across a very well-known statement that: Any two dihedral groups of degree $4$ are isomorphic. I have two questions from here: What’s actually is a ...
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### Show that the $D_{3h}$ prism is given by the direct product $D_{3h}=D_3 \otimes C_s$

I have some questions regarding the notation and solution to the following problem involving the direct product: The diagram below shows the symmetry operations of an equilateral right triangular ...
1 vote
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### Fitting length of Dihedral groups

Let $G$ be a dihedral group of order $2n$ where $n\geq 1$, denoted by $D_n$. We know that $G$ is nilpotent if and only if $n=2^i$ for all $i\geq 1$, a proof of this you can check in the below link 1. ...
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### Sylow-$2$ Subgroups of a simple group of order 168

I'm stuck somewhere in the following claim, I would appreciate if you could help: Claim: Let $G$ be a simple group of order $168(=2^3\cdot 3\cdot 7).$ Then all Sylow $2$-subgroups of $G$ are dihedral. ...
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### Consider a reflection as an element of the permutation group of the vertices of a regular (2n + 1)-gon. Is the permutation even or odd?

Let $n \ge 1$ be an integer. Let $F \in D_{(2n+1)}$ be a reflection. Consider $F$ as an element of the permutation group of the vertices of a regular $(2n + 1)$-gon. Is this permutation even or odd? ...
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### Conjugacy classes of subgroups of $D_{4}$

This problem is from Michael Artin Algebra first edition. 6.3.3) List all subgroups of the dihedral group $D_{4}$ and divide them into conjugacy classes. I am a bit unsure what I am being asked here ...
1 vote
### Intersection of two cyclic groups in $D_{n}$
For the dihedral group $D_{n}=<\rho,\tau | \rho^{n}=\tau^{2}=id, \rho\tau=\tau\rho^{-1}>$ How can we prove that the intersection of $<\rho>$ and $<\tau>$ is the trivial group $\{e\}$,...
From Aluffi's Algebra: Chapter $0$ Describe generators and relations for all dihedral groups $D_{2n}$. (Hint: Let $x$ be the reflection about a line through the center of a regular n-gon and a vertex,...