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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Showing reflections in midpoint of sides are not equal to rotations
I want to prove that the order of $D_{2n}$ is equal to $2n$. I said that there is $n$ distinct rotations ,so there must be $n$ reflection.
when $n$ is odd where n is the number of sides of regular n-...
user1357650
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Does there exist a group $G$ such that $\operatorname{Aut}(G)\cong D_5$, where $D_5$ denotes the dihedral group of order 10?
I came across this problem stated in the title having no clue what to do, and got stuck even in the finite case. Here's my attempt:
I first proved that the group $G$, if it exists, cannot be finite ...
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Dihedral Groups as Coxeter Groups
I have just completed Exercise 1.2 in the book "Combinatorics of Coxeter Groups" stated below:
Show that there exist Coxeter systems $(W,S)$ and $(W',S')$ with $|S|\neq|S'|$ such that $W\...
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Automorphism of order 2 of $D_q$
Let $q\geq 3$ be a prime number and $D_q$ be the Dihedral Group of order $2q$. Find all automorphism of $D_q$ of order $2$.
I tried this using a 'generic' automorphism $\varphi$ such that $\varphi(r)=...
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Dihedral group $D_3$(also $D_4$) - reflection compose rotation
I’ve read in a book(image link)that for dihedral group $D_3, a \circ r_1 = b $where a means reflection about $AO$ ($O$ is the centroid of an equilateral triangle $ABC$), $r_1$ means rotation about O ...
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Irreducible $2$-dimensional representation of $D_n$
Show that for $n\geq3$ the dihedral group $D_n$ has an irreducible representation of dimension $2$. Is it unique?
I thought of doing it this way: let's count the conjugacy classes of $D_n$, which are, ...
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Finding irreducible representations of $D_{2n}$ using Mackey little group method
Let $D_{2n}$ be the dihedral group on 2n elements, consisting of n rotations and n reflections. I know the group of n rotations form a normal subgroup of $D_{2n}$ and $D_{2n}$ is a semidirect product ...
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Galois Group of $x^n-p = D_n$
The question I have on my homework is to prove the following theorem.
If $n \in \{3, 4, 6\}$ and $p \in \mathbb{Z}_{>0}$ is prime, then the
Galois group of $x^n − p$ is the dihedral group $D_n$.
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What are the groups containing dihedral group $D_4$ of order $8$? [closed]
I'm a little embarrassed to ask this but I couldn't answer it myself.
I am looking the groups that contains $D_4$ and larger than $D_4$. Here is what I think:
We cannot say $D_4 \subseteq D_n, n\ge 5$ ...
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How to prove the enhanced power graph of dihedral groups?
According to Aalipour et al. (2016), they defined the enhanced power graph of group $G$ as a simple undirected graph where the vertices are all elements of $G$ and two vertices, $x$ and $y$, are ...
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How can we distinguish elements of $D_n$ that include reflections versus those that don't?
For $n \geq 3$, the dihedral group $D_n =\langle r, s \rangle$, where $r ^n = s^2 = e$.
Within this group, we can distinguish two types of elements:
Those of the form $r^i$, where $i$ is any integer
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Proving the isomorphism type of the commutator subgroup of the dihedral group $D_n$
For any $n$, to what group is the commutator subgroup of the dihedral group $D_n$ isomorphic to?
My solution is below. I request verification, feedback, and improvements. In particular, can you help ...
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Counting subgroups of dihedral groups
Question:
How many subgroups of order 6 does $D_6$ have? How many does $D_{12}$ have? Generalize to $D_n$ where n is a positive integer divisible by 6.
Here, order of $D_n = 2n$.
Please don't use ...
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How does $ab= a^2ba$ for the Dihedral group $D_8 = \langle a, b \rangle$ and a $\mathbb{C}D_8$-Algebra?
I'm trying to show for $z = b + a^2b \in \mathbb{C}D_8$, $az = za$.
I have the presentation $D_8 = \langle a, b \ : \ a^4 = 1, b^2 = 1, b^{-1}ab = a^{-1} \rangle$. So I want to show $az = ab + a^3b = ...
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Arithmetic on elements of $D_{2n}$
I've seen the arithmetic on the dihedral group $D_{2n}$ written in several different ways, but here's what I'm working with. Say that $r$ is a clockwise rotation by $\frac{2\pi}{n}$ radians and $s$ a ...
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