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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Terminology for dihedral groups

What notation is most common for the dihedral group of order $2n$? I'm talking about the group of symmetries of a regular $n$-gon. I know that some books call this group $D_n$, and some books call it $...
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51 views

Show that $D_3\times_\rho\mathbb{Z}_2$ is not isomorphic to $A_4$

This is the third part of a problem. I will list here the first two parts as a reference and then my attempt to solve the third one. I want to verify if the first and second are right and some help ...
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1answer
521 views

Number of conjugacy classes of a Dihedral group?

How do you find the number of conjugacy classes of a Dihedral group? Say for D11 for example. I know by Lagrange each conjugacy class has order 1, 2, or 11. For smaller n, it can sometimes just be ...
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161 views

Number of group homomorphisms from infinite cyclic group to dihedral?

Want to determine the number of group homomorphisms $f: \mathbb{Z} \to D_7$. My guess is that there is only $1$ because $0$ is the only element with finite order in $\mathbb{Z}$. Note a cyclic group ...
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118 views

Are dihedral groups indecomposable?

This is Exercise 6.29 from An Introduction to the Theory of Groups by J. J. Rotman: Show that the following groups are indecomposable: $\mathbb{Z}$; $\mathbb{Z}^{p^n}$; $\mathbb{Q}$; $S_n$; $D_{2n}$...
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1answer
31 views

In Dihedral group $D_n$ write the following in the form $r^i$ or $r^i f$ where $0\leq i < n$

In Dihedral group $D_n$ write the following in the form $r^i$ or $r^i f$ where $0\leq i < n$. (r is rotation and f is reflection, I think) Also given, $r=R_{360/n}$ a) In $D_4$, $fr^{-2}fr^5$ b)...
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1answer
431 views

Is the Dihedral Group $D_5$ solvable? [duplicate]

I just showed that $D_4$ is solvable since it is isomorphic to $\mathbb{Z}_2 x \mathbb{Z}_2$ so since $\mathbb{Z}_2$ is solvable so is $D_4$. Does something similar extend to $D_5$ or is there a ...
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80 views

Matrix representation of $D_5$ rotation in $4D$

I am working with a $4$-dimensional representation of $D_5$. The $4D$ representations of the generators of $D_5$, $r$ (a rotation by $\pi/5$) and $p$ (the reflection $x \rightarrow x, y \rightarrow y$...
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1answer
102 views

Non-orthogonal matrix representation of dihedral groups

I found this table on here: $a$ is the first element of a dihedral group $D_n$, i.e. the rotation by an angle $2\pi/n$. Few questions on that: 1) Non real, non orthogonal representation matrices. ...
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1answer
199 views

Dihedral Group of Order $2^{n}$ where $n \geq 3$

As I was self studying finite group theory I noticed something intriguing and failed to provide proof for the claim. What I noticed was that the order of the Dihedral group of order $2^{n}$ where $n \...
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164 views

Show every irreducible representation of $D_{n}$ must have dimension less than or equal to 2

This question was homework once upon a time. I have long since handed it in. "Let $D_{n}$ be the dihedral group with $2n$ elements. Show that every irreducible representation of $D_{n}$ must have ...
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203 views

Subgroup of Dn isomorphic to Q8

Is there an n such that $D_n$ contains a subgroup isomorphic to $Q_8$? My immediate thought is no, but I'm not sure how to prove it. I know that there are only 2 non-Abelian groups of order 8 (up to ...
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2answers
38 views

Relation between group algebra, representation and companion matrix

Padron my ignorance, I am a physicist who never really fully understood group theory but just used it when necessary. Let's consider a $C_5$ group corresponding to $5$-fold rotation by $\frac{2\pi}{5}...
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249 views

Confusion about centraliser of $D_5$ in $GL_4(\mathbb{Z})$

I am trying to follow a derivation on a very old paper. My knowledge of group theory is limited, I have the basis but not much experience with advanced concepts. We are working in 4 dimensions, so ...
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49 views

Find the centraliser of $D_5$ in $GL_4 (\mathbb{R})$

I want to find the centraliser of $D_5$ in $GL_4 (\mathbb{R})$. I can find the centres of $D_5$, i.e. the subgroups of elements that commute with a particular element just by playing around with its ...
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1answer
295 views

Determining order of elements and number of automorphisms in dihedral groups

Reviewing some stuff and found myself confused at a few things involving dihedral groups and automorphisms, would very much appreciate some assistance in understanding. Namely beginning with this, I ...
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466 views

Orbits of rotations under action by D4.

If D4 is acting on the subgroup of its rotations, C4, by conjugation, what are the orbits? I believe that the orbit of each rotation is itself and its own inverse rotation and nothing else. For the ...
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75 views

Determine whether pairs of elements belong to the same left H-cosets

$ G = D_6$ (dihedral of order 12, with generator $a$, $b$ where $a^6 = b^2 = e$ and $ba=a^{6−1}b$ $H=⟨a^1b^0⟩ $ Given $(x,y)$ pairs, $(a,b^0), (b,a^5b), (b^0a^1, a^4)$, determine if $xH = yH$. The ...
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1answer
197 views

Describe all homomorphic images of the dihedral group $D_3$ of order 6

I saw a solution for $D_4$ where they just found all the normal subgroups and found the quotient groups. Is that sufficient here? How do I even do that? I know that one of them is the kernel of $D_3$...
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1answer
299 views

Compute cosets in the dihedral group

Stuck on how to answer this. So I understand the dihedral group $D_8$ consists of $16$ elements with $n$ rotations and $n$ reflections. Each of the reflections have order $2$ so $b^2 = 1$ as shown. $...
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Show $T(x)=x+1$ and $I(x)=-x$ produce a dihedral group$G := D_n$

Lets look at a musical note system with $n$ notes. We see two notes as the same when they differ one octave. We write the collection of notes as $X= \Bbb Z_n$ $T: X \rightarrow X$, $T(x)=x+1$ ...
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291 views

Defining dihedral groups $\{\sigma \in S_n: $ something $\}$

I am trying to understand hos one can define the dihedral groups $D_n$. I have seen the "definition" that just says this is the group of symmetries of an $n$-polygon. So you have rotations and ...
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Tricky question on Dihedral groups [duplicate]

If $$D_{2n} = \{r^i s^j, \quad 0 ≤ i ≤ n − 1, \quad 0 ≤ j ≤ 1 \}$$ is the dihedral group of order $2n$, show that the center of $D_{2n}$, $n > 2$ is the trivial subgroup if $n$ is odd, and is the ...
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39 views

Show that the dihedral group $D_6$ of order $12$ has a nonidentity element $z$ such that $zg = gz$ for all $g ε D_6$.

From notes, I think all of the following are true: Every element of $D_6$ can be written as $s^ir^j$, where $i = 0,1$ and $0\le j\le 5$. $r^6 = e$, where $e$ is the identity. $s^2 = e$ $r^ks = sr^{-k}...
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2answers
78 views

$D_{12}$ is isomorphic to $\Bbb Z_3\times D_4.$

Prove/Disprove: $D_{12}$ is isomorphic to $\Bbb Z_3\oplus D_4.$ $D_{12}$ has 11 elements of order $12$ whereas $\Bbb Z_3\oplus D_4$ has 9 elements of order $12$ since $\Bbb Z_3$ has $3$ ...
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1answer
49 views

Need help with a proof involving subgroups of $D_{n}$

I have a question that needs attention regarding the dihedral group $D_{n}$. Here is the context of the problem: Consider the Dihedral group $D_{n}$. Let $\sigma$ be a rotation counterclockwise by $\...
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What does “A function on the $n-$gon” mean?

I am reading Dummit and Foote. We have: For each $n \in \mathbb{Z}^+, n \ge 3$ 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 ...
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3answers
367 views

Dihedral Groups; what exactly are the elements of the set?

I am reading Dummit and Foote. We have: [Definition 1:]For each $n \in \mathbb{Z}^+, n \ge 3$ let $D_{2n}$ be the set of symmetries of a regular $n-$gon, where a symmetry is any rigid motion of the ...
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1answer
123 views

Expressing dihedral group as internal direct product of normal subgroups

Can dihedral groups be expressed as internal direct product of two normal subgroups? I think no, since an element of a normal subgroup must commute with every other element of other normal subgroup, ...
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1answer
159 views

Why are Transpositions and rotations in dihedral groups the same operation?

A dihedral group seems to have 2 operations, rotation, and transposition. These transformations seem so distinct, no composition of rotations may ever lead to a transposition and vice versa (right?). ...
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3answers
161 views

Help with understanding proof: every group of order 6 is cclic or dihedral.

I am considering the proof that every group of order 6 is cyclic or dihedral (following some lecture notes). The initial part of the proof, in broad brushstrokes, considers that the elements in the ...
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1answer
87 views

Showing that the dihedral group $D_n = \{a^i b^j \mid 0 <i <6, \ 0 <j< 2 \}$ quotiented by $\langle a^k \rangle$ is isomorphic to $D_k$

Consider the dihedral group $D_n = \{a^i b^j \mid 0 \leq i <6, \ 0 \leq j< 2 \}$ where $a^n = b^2 = 1$ , and $a b =b a ^{-1}$. For any divisor $k$ of $n$ show that $$\langle a^k\rangle\...
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93 views

What are the left cosets of $ D_{8} $ with respect to the subgroup $H=\langle a^2\rangle$?

Let $D_8$ denote the dihedral group of order $16$, aka the group of symmetries of the regular $8$-gon. Using Lagrange's theorem there are $16/4$ cosets which I have worked out to be $H$, $aH$, $bH$, $...
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1answer
427 views

Dihedral Group of Order 12 in Cycle Notation

I want to check that this cycle notation is correct for the Dihedral Group of order $12$. I found this graph in Wikipedia. I did not find the cycle notation. May you please tell me if my cycle ...
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2answers
326 views

Show that the additive group $\mathbb R$ acts on the $x, y$ plane $\mathbb R \times \mathbb R$ by $r\cdot (x, y)=(x+ry,y)$.

Show that the additive group $\mathbb R$ acts on the $x, y$ plane $\mathbb R \times \mathbb R$ by $r\cdot (x, y)=(x+ry,y)$. I am completely lost with this one partly because I do not understand group ...
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1answer
125 views

Direct product of $D2$ and $D3$

I am trying to find information on the group resulting from the direct product of the dihedral groups $D2$ (Klein four-group) and $D3$ (or, isomorphic: $S_3$ or $C_{3v}$). What would be the resulting ...
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1answer
40 views

Showing $r^{n/2} \in D_{2n}$ commutes with $f$ (where $n \equiv 0 \text { mod }2$)

This is part of a bigger problem, but I just wanted to ensure the work I did in this portion is correct. I'm using the generators $r$ and $f$, where $r^n=e$, $f^2=e$, and $rf=fr^{n-1}$. Here's my ...
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1answer
110 views

Determining the number of conjugacy classes of $D_{2n}$ for $n$ even and for $n$ odd

How would one go about finding the precise number of conjugacy classes for any given $n$ if $n$ is odd or even? By looking at the cases of $D_6$, $D_8$ and $D_{10}$ I have some idea, but I'm not ...
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64 views

Finding Homomorphisms

Let D be the dihedral group of order 6 and C be the cyclic group of order 6. Find all homomorphisms D $\longrightarrow$C, (you may assume standard facts about cyclic and dihedral groups but you must ...
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227 views

If $ |G|=2 p$ then either $G$ is cyclic or $G \cong D_{2\cdot p}$

I'm studying for my Groups and Rings exam and I came up with this exercise which I'm having some struggle with: Let $G$ be a finite group of order $2 p$ ($p>2$ with $p$ prime). Proof that either $...
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208 views

How many reflection subgroups are in $D_{2n}$?

Given the dihedral group $D_{2n}$ of order $2n$, is there a formula for the number of reflection subgroups of $D_{2n}$?
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107 views

Action of $D_{2n}$ on pairs of opposite vertices of a regular n-gon

Having some difficulty understanding the solution to this problem. We have an action of $D_{2n}$ on set $A=\left\{{\left\{\overline{a},\overline{k+a}\right\}\mid1\leq a\leq k}\right\}$ and $s.\left\...
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3answers
1k views

How to show ${D}_3$ is not cyclic?

I understand how to show if a group involving $\mathbb{Z}$ is cyclic but not in the case of dihedral groups. I am specifically interested in showing (or knowing) that $D_3$ is not cyclic. How do I go ...
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1answer
43 views

Dihedral Group $D_{2016}$. Simplify $a^3ba^2baba^{-1}$ in terms of $b^ra^m$, where r and m are non negative integers.

We know that $a^{1008}=b^2=e$ and $a^jb=ba^{-j}$ for all $j$ I keep getting the answer $ba^{-3}$ but $-3<0$ so it is negative. How should I do this properly?
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1answer
159 views

Group of Rotations Stabilizer in D4

In D4, the subgroup of rotations is not a stabilizer for any point in a square (even the center). Am I missing anything? Thanks! Edit: To clarify, I wanted to ask if the subgroup of rotations of D4 ...
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1answer
550 views

Understanding the dihedral group

I know that the dihedral group is $$D_{2n} = \{1, r, r^2, \dotsc, r^{n-1}, s, sr, \dotsc, sr^n-1\}$$ where the $r^i$ are rotations and $s$ is a symmetry. Now, what I want to know is what is the ...
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1answer
72 views

Check my logic: Does a generalized dihedral extension necessarily contain the extension element? (I say yes.)

I want to make sure I am correct about something I read in Moore and Pollatsek's "Difference Sets": Suppose we have an abelian group $H$. If $\exists g\notin H$ with $g^2=1$ and $ghg^{-1}=h^{-1}\;...
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2answers
78 views

Dihedral groups and reflections

Let $\rho\in D_{10} $ be a rotational symmetry in $\frac{2\pi}{5}$ radians non-clockwise,and let $\epsilon\in D_{10}$ be a reflection symmetry (related to the X-axis).Prove that every $\epsilon\rho^i\...
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4answers
1k views

Some Subgroup of Dihedral Group is Normal

I ran into this question when I was studying for my abstract algebra midterm. Show that the subgroup $H$ of rotations is normal in the dihedral group $D_n$. Find the quotient group $D_n/H$. I'm ...
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1answer
121 views

General conditions on generating sets for dihedral groups

It's well-known that, for any $n$, we can consider $D_n = \langle r, f | r^n = e, f^2 = e, r^k \not = e (0 < k < n), fr= r^{-1} f\rangle$; However, not all two element generating sets for $...