# Questions tagged [dihedral-groups]

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

742 questions
Filter by
Sorted by
Tagged with
25 views

### 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-...
188 views

### 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 ...
• 553
1 vote
37 views

61 views

### 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 ...
75 views

### 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, ...
1 vote
29 views

### 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 ...
85 views

### 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 $D_n$. ...
• 811
81 views

### 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 $D_4 \subseteq D_n, n\ge 5$ ...
• 193
1 vote
15 views

### 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 ...
1 vote
37 views

### 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 ...
• 4,450
1 vote
81 views

### 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 ...
• 4,450
51 views

### 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 ...
• 47
68 views

35 views

### Automorphisms of the Infinite Dihedral Group

Let $D_\infty$ be generated by $a$ the reflection fixing the line $\{x=0\}$ and $b$ the reflection fixing the line $\{x=1\}$ in $\mathbb{R}^2$. I want to show that any $\alpha\in \text{Aut}(D_\infty)$ ...
249 views

### A question regarding the dihedral group $D_4$.

I know that the dihedral group $D_4$ is generated by two elements $\sigma$ and $\tau$ such that $\sigma^4=\tau^2=e$, and $\tau\sigma\tau^{-1}=\sigma^3$. This is essentially the group of eight elements ...
89 views

• 915
1 vote
91 views

### $H,K \leq G$, $|H| = 126, |K| = 228$. Show that $H \cap K \leq G$ is either cyclic or isomorphic to $D_3$

So, I don't have to prove that $H \cap K$ is a subgroup of $G$, and $G$ is finite. I know that any group of order $6$ is isomorphic to either $D_3$ or $\mathbb{Z}_6$ (which is cyclic) so I'm assuming ...
• 456
118 views

• 3,047
142 views

• 785
309 views

Definition: Here is equivalent definition of dihedral group \begin{align}D_{2n}&=\{\sigma \in S_n\mid \sigma(a+1)\equiv\sigma(a)+1\pmod n, \forall 1\leq a\leq n\} \bigcup \{\sigma \in S_n\mid \... • 4,249 0 votes 2 answers 90 views ### Help understanding how to show that two groups are isomorphic I'm trying to understand what is required to show that two groups are isomorphic. My understanding is that two groups G_1 and G_2 are isomorphic if:  |G_1| = |G_2|  (equal cardinality) G_1 ... 1 vote 1 answer 292 views ### Definition of Dihedral Group in Dummit’s Abstract Algebra For each n\in \Bbb{Z}^+, n\geq 3 let D_{2n} be the set of symmetries of a regular n-gon, where a symmetry is any rigid motion of the n-gon which can be effected by taking a copy of the n-... • 4,249 4 votes 0 answers 114 views ### Geometrical interpretation of the faithful actions of D_6 and D_{10} on sets of size 5 and 7, respectively. [duplicate] The dihedral group of order 2n, D_n, acts faithfully on the set of the n vertices of the regular n-gon. For n=6 and n=10, D_n acts faithfully also on sets of size smaller than n, ... • 3,393 2 votes 3 answers 75 views ### The dihedral group D_n cannot be written as a product of two nontrivial groups Let n be an odd prime. I want to prove that the dihedral group D_n cannot be written as a product of two nontrivial groups. But it seems problematic, as I completely neglect the properties of ... 0 votes 0 answers 49 views ### Understanding cycle notation for a dihedral group and one of its subgroups I'm looking at an example that instructs me to consider the group: G = \langle(1, 2, 3, 4, 5, 6, 7, 8), (1, 8)(2, 7)(3, 6)(4, 5)\rangle \cong Dih(16) $$And the subgroup:$$ H = \langle(1, 3, 5, 7)(...
• 67
I need to list all the subgroups of the symmetry group of the cube that are isomorphic to either $\mathbb{Z}_n$ or $\mathbb{D}_n$ (the dihedral group) for some $n \in \mathbb{N}$. I know that the ...