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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6
votes
2answers
41 views

Finding $N(D_{4})/D_{4}$ for $D_{4}$ in $D_{16}$

I want to find $N(D_{4})/D_{4}$ where $N(D_{4})$ is the normalizer of $D_{4}$ in $D_{16}$. I'm not too clear on what the normalizer of $D_{4}$ in $D_{16}$ Is there a nice way to find $N(D_{4})/D_{4}$?...
3
votes
1answer
2k views

How to write dihedral group in cycle notation?

Since each symmetry can be thought of as a permutation of the vertices, the elements of $D_n$ can be thought of as elements of $S_n$. So I'm wondering if there's a systematic way that we can always ...
1
vote
0answers
19 views

Interpreting $S_{N} \wr D_{m}$

I am trying to interpret $S_{N} \wr D_{m}$ in the light of the interpretation of $\mathbb{Z}^n_2 \wr \mathbb{Z}_2$ in this paper. So, according to the definition, $$\mathbb{Z}^n_2 \wr \mathbb{Z}_2 =...
3
votes
3answers
56 views

Let $n \ge 3$. Prove that there are $n!$ different one-to-one homomorphisms from $D_n$ to $S_n$

Let $n \ge 3$. Prove that there are $n!$ different one-to-one homomorphisms from $D_n$ to $S_n$. I know there are $n!$ elements in $S_n$, but this fact didn't get me anywhere. I tried many things ...
1
vote
1answer
200 views

How do I find all of the orbits and stabilisers of X?

Consider $D_{10}$ The group of symmetries of the regular pentagon. Let $\sigma= (12345)$ and $\tau =(13)(45)$ being rotation by $72^{\circ}$ and reflection (with 2 being the fixed point) respectively. ...
0
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2answers
124 views

Group Theory: How to find all possible images of $ f $?

Let $H$ be a group and suppose that $ f: D_{10} \rightarrow H $ is a homomorphism. How do I describe and justify all the possible images of $f$. $D_{10} = ({1, \sigma, \sigma^2, \sigma^3, \sigma^4, \...
4
votes
1answer
325 views

Why is this group of matrices isomorphic to the dihedral group?

I am reading Abstract Algebra, Theory and Applications by Judson and in exercise $13$ chapter $9$, Isomorphisms, I need to prove that the set of matrices $$A=\pmatrix{ \omega & 0 \\ 0 & \omega ...
2
votes
1answer
75 views

About proving that $\operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n)$ [closed]

How can I prove that $$ \operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n), $$ where $\mathbb {D}_n$ is the dihedral group. Can someone help me please? ...
2
votes
1answer
65 views

Is there a surjective homomorphism from $D_{10}$ to $\mathbb Z _2\times \mathbb Z_2$

Is there a homomorphism $f\colon D_{10} \to \mathbb Z_2\times \mathbb Z_2$ that is onto? Attempt: $D_{10} = \{e,s,r,...,r^9,sr,...,sr^9\}$, $\mathbb Z_2\times \mathbb Z_2 = \{(0,0),(1,0),(0,1),(1,1)\...
1
vote
2answers
3k views

Show the normal subgroups and cosets of a dihedral group (D6)

$G=D_6$ and $H=<R^2>$. Use this Cayley table for $D_6$ (a). Show that $H \vartriangleleft G$. I want to show by finding out $aH=Ha$ for all $a \in G$, but then how do I proceed, it would be ...
0
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0answers
624 views

Is $D_{2n}$ isomorphic to $D_n \times \Bbb{Z}_2$ for all $n$? For all odd $n$? [duplicate]

Is $D_{2n}$ isomorphic to $D_n \times \Bbb{Z}_2$ for all $n$? For all odd $n$? I just want to see if my thinking is sound here. My thought process is this. $\mathbb{Z}_2 \cong \{e,j\} \subset D_n$ ...
0
votes
1answer
372 views

Kernel of a homomorphism (dihedral group)

Suppose that $f: D_{18} \to GL(2,\Bbb R)$ is a homomorphism, $\lvert r \rvert = 18$ and $f(r) = R := \begin{pmatrix}1& 1 \\ -1& 0\end{pmatrix}$. What is $\lvert \ker(f) \rvert$? Attempt: I ...
0
votes
1answer
57 views

Symmetric Group and Dihedral Group relationship

Show that $D_3 = S_3$ but $D_n \subsetneq S_n$ for $n \geq 4$. In my class, we proved that $D_n$ is generated by $f = (1,2,3,...,n)$ and $g=(1)(2,n)(3,n-1)...(\frac{n+1}{2}, \frac{n+3}{2})$ (for an ...
1
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1answer
83 views

When is the $k/n$ representation of $D_n$ irreducible, and why?

The $k/n$ representation of the Dihedral group of order $2n$ in $GL(2,\mathbb{C})$ is induced by mapping the rotation element of $D_n$ to the Rotation Matrix $R(\frac{2\pi k}{n})$, and the reflection ...
1
vote
1answer
45 views

Are dihedral groups well defined by their generating groups?

It is well known that $D_n=<r,s | r^n=s^2=id, srs=r^{-1}>$. Now, given a group of 2n elements and a generating group of two elements satisfying the above relations is it isomorphic to $D_n$?
1
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0answers
29 views

What exactly is a Frieze group and how would you find the isometries preserving one?

So far, I've come across several examples of frieze groups, but I've not yet come across an understandable definition of what they are. I've also been asked questions that ask me to state the ...
0
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1answer
318 views

Show the sub group $\langle r^k\rangle$ is normal in the dihedral group of order $2n$ when $k | n$ and $r$ is the rotation by $2\pi/n$

let $ k | n ; n,k \in \Bbb Z $ let $r$ be the rotation in the plane by an angle $2 \pi \over n$ prove the subgroup $ \langle r^k\rangle $ of $D_n$ is normal. Further is there a normal subgroup ...
1
vote
1answer
27 views

The order deduced from relations in $D_n$

If $D_n \triangleq \langle a,b | a^n=e, b^2=e, abab=e \rangle$, can it be proved that the order of $a, b$ is actually $n$ and $2$ respectively ? I mean can the relations on the right somehow after ...
1
vote
1answer
47 views

$D_n$ is a group for all integers $n\geq 3$

If one wants to prove that $D_n$ (a dihedral structure) is a group for all $n\geq 3$, would it be sufficient to state the following? $D_n = \{1, s, r, ..., r^n, sr, ..., sr^n\}$. Associativity: $(1s)...
1
vote
1answer
198 views

Suppose $F_1$ and $F_2$ are distinct reflections in $D_n$ such that $F_1F_2=F_2F_1$…

Suppose $F_1$ and $F_2$ are distinct reflections in $D_n$ such that $F_1$$F_2=F_2F_1$, prove that $F_1F_2=R_{180}$. I'm stumped on where to even start. Up to the point where I've gotten the book I am ...
0
votes
0answers
43 views

Automorphism group of disjoint cycle graphs of different lengths

This question is supplementary to another question. From that question, we know that the automorphism group of the $N$ disjoint cycle graphs of same length $n$ is $S_N \wr D_n$. My question: What is ...
2
votes
1answer
137 views

Group isomorphism between $D_3$ and $S_3$

If one wants to prove that $D_3$ is isomorphic to $S_3$, would it be sufficient to define a homomorphism $\psi: D_3\to S_3$ and argue that it is well-defined since $\psi(sr^i)=\psi(s)\psi(r)^i=\...
2
votes
1answer
104 views

Find all homomorphisms from $D_{2n}$ to $\mathbb C^\times$ (revisit)

I actually was asking the same question in here but haven't gotten any feedback yet. I now can elaborate a little so that final answer would be closer. I wanted to find all homomorphisms from the ...
2
votes
2answers
295 views

Conjugacy classes for rotations of $D_{2n}$

It says in my notes that for a dihedral group $D_{2n}$, if $n$ is even, then each conjugacy class has at most $1$ element. It says that for the conjugacy class $C_{r^k}$ where $r^k$ is some reflection,...
1
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0answers
351 views

Number of $p$-Sylow subgroups in $D_{2n}$

Let $2n = 2^ak$, where $k$ is odd. We wish to show that the number of $2$-Sylow subgroups of $D_{2n}$ is $k$. My approach has been to construct such a $2$-Sylow subgroup $P_2$, and then show that the ...
7
votes
1answer
5k views

Center of dihedral group

I am trying to solve the following exercise about the dihedral group and its center: If $g\in Z(D_{2n})\Leftrightarrow ga=ag, bg=gb$, where $a,b$ are generators of $D_{2n}$. We have defined the ...
9
votes
1answer
999 views

Finding Sylow 2-subgroups of the dihedral group $D_n$

I am trying to describe the Sylow $2$-subgroups of an arbitrary dihedral group $D_n$ of order $2n$. In the case that $n$ is odd, $2$ is the highest power dividing $2n$, so that all Sylow $2$-...
0
votes
1answer
52 views

Commutativity in dihedral group of order 2n

How does one show that $r^{n/2} \in D_{2n}$ commutes with $sr^i$? Here's what I've tried: Showed that $r^{n/2}$ commutes with $s$. For $sr^i$, for integer $i \in [1, n-1]$, I tried this: $sr^i r^{n/2}...
1
vote
0answers
225 views

Describing all Sylow 2-subgroups of the dihedral group $D_n$

We are trying to find all Sylow 2-subgroups of an arbitrary dihedral group $D_n$ of order $2n$. In the case that $n$ is odd, any Sylow 2-subgroup must have order 2, and it is fairly easy to deduce ...
0
votes
2answers
22 views

If $|G|=2n$, $|a|=n$, $|t|=2$ and $tat=a^{-1}$ then $t\not \in \langle a\rangle$

Let $G$ be a group of order $2n$ and $a,t\in G$ s.t $|a|=n, |t|=2$ and $tat=a^{-1}$. define $N=\langle a\rangle$. I need to show that $|G:N|=2$ and $G=N\cup tN$. to show that $|G:N|=2$ is trivial ...
1
vote
2answers
513 views

Generators of a Dihedral Group

Let $D_4=\{ 1,r,r^2,r^3,s, sr, sr^2, sr^3\}$. I want to show that $<s> $ is a normal subgroup of $<s,r^2>$ but $<s>$ is not a normal subgroup of $D_{4}$. I think $<s> = \{1,s\}...
2
votes
1answer
221 views

Show that there exist an injective homomorphism of dihedral group $D_n$ into $G$.

Let: $$n\gt2 \; \text{and the group} \; (G,⋅)$$ Consider that there existe: $a,b∈G$ such that $a^n=b^2=1_G$ and $b⋅a=a^{−1}⋅b$ and $n$ is the smallest $n≥1$, such that $a^n=1_G$. ...
1
vote
2answers
132 views

Element not in the subgroup $<x>$ of the dihedral group is a reflection

The dihedral group $D_{2n}$ is generated by $x$ and $y$ such that $x^n = y^2 = xyxy = e$. Show (algebraically) that elements not in the subgroup $<x>$ is a reflection and find the line (...
3
votes
3answers
721 views

Are all abelian subgroups of a dihedral group cyclic?

Are all abelian subgroups of a dihedral group cyclic? Attempt: I have counter-examples for n=1,2 so I know that it isn't true for n<3. Is it true for n≥3? How do you know this?
2
votes
3answers
132 views

$D_3\oplus D_4$ not isomorphic to $D_{24}$

We need to prove that $D_{3} \oplus D_{4}$ is not isomorphic to $D_{24}$ . The way in which I approach such type of questions is to count the number of elements of order $x$ in one group and then in ...
1
vote
2answers
202 views

Group Theory- symmetry group

The Symmetric group of the set ${1,2...,n}$ is $S_n$. It is the set of permuations of the set ${1,2...,n}$. But what is the symmetry group of a polygon? or a $3D$ shape? For example I saw: "A ...
-3
votes
1answer
830 views

A group is generated by two elements of order $2$ is infinite and non-abelian

My question is as follows: Let $G = \langle a,b \mid a^2=b^2=1 \rangle $ be a group generated elements $a, b$ and the equation $a^2=b^2=1$. Prove that $G$ is infinite and non-abelian. I got the ...
2
votes
1answer
47 views

How shall we show that the only possible orders of any element in dihedral group $D_n$ will be either a divisor of 2 or $n$?

I am studying the dihedral group $D_n:=\{r_n, f_n: r_n^n=f_n^2=(r_nf_n)^2=e_n\}$. I am willing to show that the possible orders of any element in it will be either a divisor of 2 or $n$. But don't ...
3
votes
1answer
209 views

Show $D_{2n} \to GL_2(\mathbb{R})$ is an injective homomorphism

Show $\phi: D_{2n} \to GL_2(\mathbb{R})$ is an injective homomorphism, where $\phi: \textrm{rotations} \to \left( \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{...
3
votes
3answers
1k views

Normalizer of a Sylow 2-subgroups of dihedral groups

I can't solve the following exercise which is the last exercise in page 146 of Dummi & Foote's Abstract Algebra: Let $2n=2^ak$ where $k$ is odd. Prove that the number of Sylow 2-subgroups of $...
2
votes
0answers
69 views

How can I show that $D_{2n}$ follows from these relations?

Suppose we have a group $A$ which is generated by generators $R$ and $F$, subject to the relation $$ R^n=I, F^2=I,RF= FR^{-1}.$$ It should be just the dihedral group of order $2n$, the one generated ...
3
votes
3answers
277 views

Why Composition and Dihedral Group have reverse order of operation?

NOTE - I didn't receive any answer in here and I think because my first post is not clear, so I entirely made another example: $K={\{id,r^2,r^4,s,r^2s,r^4s}\}$ is a proper subgroup of the dihedral ...
1
vote
0answers
37 views

$x,y \in G$, $o(x) = 2, o(y) = 2, o(xy) = k, k\geq 3$. Show that $D_k \cong \langle x,y\rangle$

I know that $D_k = \{1,r, \dots, r^{k-1}, s, sr, \dots, sr^{k-1} \}$ and that I can use $\phi: s^i r^j \mapsto x^i(xy)^j$. I don't know what can be found using this.
1
vote
1answer
79 views

Dihedral Group question involving $D_{4n}$

Let $D_{4n}$ be the dihedral group of the order $4n$. Prove that $D_{4n}/\langle T^n \rangle$ is isomorphic to $D_{2n}$. We tried to configure an action on the diagonals of the $n$-gon and prove ...
3
votes
2answers
76 views

Dihedral subgroups of $S_4$

Prove that in $S_4$ there are $3$ groups that are isomorphic to $D_4$. I know that the $2$-sylows of $S_4$ should be subgroups of order $8$, but to prove it is a bit tricky for me Any help would be ...
0
votes
1answer
48 views

Dihederal Group $D_{2n}$ Where $n$ is even/odd

I know that the group presentation of $D_{2n}$ is the following $$D_{2n} = \big<a,b: a^n=b^2=1,b^{-1}ab =a^{-1} \big>$$ Now if we consider the case where $n$ is even and we write $n =2m$ for ...
1
vote
0answers
334 views

I need to prove that there is one homomorphism $\varphi : Dn \to Sn$ such that

I need to prove that there is one homomorphism $\varphi : Dn \to Sn$ such that $\varphi$($\tau$) = $$ \begin{pmatrix} 1 & 2 & . & .& . & n \\ 2 & 3 &...
2
votes
2answers
296 views

Is the infinite dihedral group an inverse limit of the finite dihedral groups?

The p-adic numbers are the inverse limit of the rings $\mathbb Z / p^n \mathbb Z$. Can the infinite dihedral groups be construed as some sort of inverse limit of finite dihedral groups?
3
votes
2answers
202 views

Elements of Dihedral Groups

Is there an elegant way of showing that the elements of a dihedral group are only rotations and reflections? Specifically, I'm having trouble convincing myself that a composition of a rotation and a ...
1
vote
1answer
79 views

Finding subgroups of $D_{12}$

Let $G=D_{12}=\langle a,b \mid a^6=b^2=e, bab^{-1}=a^{-1} \rangle$. Find all subgroups of $G$. We can easily spot the cyclic normal subgroup $C=\langle a \rangle = \{e,a,a^2,\dots,a^5\}$. Now $C$ ...