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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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
201 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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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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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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1answer
94 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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126 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
52 views

Writing Dihedral Groups in Canonical Form

For each of the following elements in $D_{14}$, write in canonical form: $x^6x^5yx^3xyyyx^{−2}$ $(x^2yx^{−3})^2 $ $(yx^{-7}x^2yyx^4 )^{−1}$ I've looked everywhere online and in my text book and have ...
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42 views

Smallest symmetric group which embeds $D_n \rtimes Aut(D_n)$

This question answers which the smallest symmetric group to embeds $D_n$ of order $2 n$. I would like to know what is the smallest symmetric group which embeds $D_n \rtimes Aut(D_n)$. I understand ...
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Visualizing $S_3 \rtimes D_4$

I am trying to visualizing $S_3 \rtimes D_4$ following this video. Here, $S_3$ is the symmetric group over three symbols and $D_4$ is the dihedral group of order $8$. The semidirect product is defined ...
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1answer
156 views

Is $D_{∞}^{(2)}$ virtually abelian?

I know that $D_{∞}^{(2)}$ is the set of all bijections: $f: \mathbb{Z}^{2}\rightarrow \mathbb{Z}^{2}$ such that $d(f(x),f(y))=d(x,y)$. So the group multiplication is given by the composition of ...
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193 views

$D_{2n}$ Acting on Opposite Pairs of Vertices

I have been asked to show that $D_{2n}$ acts on the set consisting of pairs of opposite vertices of a regular $n$-gon, where $n$ is taken to be a positive even integer. The first thing that came to ...
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1answer
17 views

Find the solution to $x$ in $D_4$ to $x^2 = r_0$

I attempt to solve $x^2 = r_0$. Things I know $r_i ∘ r_j =$ $r_{i+_nj}$ $r_i ∘ s_j =$ $s_{i+_nj}$ $s_i ∘ r_j =$ $s_{i-_nj}$ $s_i ∘ s_j =$ $r_{i-_nj}$ I know how to solve the linear type, I am ...
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1answer
39 views

Showing that $\phi$ is a homomorphism

$G=\mathbb{Z}^2$ is a group with product $(a,b)\cdot(c,d)=(a+c,(-1)^cb+d)$. Show that the image $\phi: G \to D_{10}$ with $(a,b) \mapsto s^ar^b$ is a homomorphism ($D_{10}$ is the dihedral group of ...
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2answers
54 views

Finding groups $H$ for which there exists surjective homomorphisms $f:D_4 \rightarrow H$?

How can I find out for which groups $H$ there exists surjective homomorphisms $f: D_4 \rightarrow H$? $D_4$ is the dihedral group of the square. I have a theorem that says that there exists such ...
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76 views

Computing the characters of $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$

How can I compute the characters of the induced representation $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$? Here, $S_n$ is the symmetric group over $n$ symbols and $D_n$ is the dihedral group of order $2 ...
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45 views

Row in the character table of $D_{10}$

Give the values of one row of the character table of $D_{10}$ corresponding to a character of degree $2$ I know the conjugacy classes of $D_{10}$, the dimensions of the irreducible representations ...
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610 views

Find the normal and the composition series

Could you give me some hints how we could find a normal series and all the composition series of $D_4$ ? $$$$ A normal series of $G$ is $$G\geq G\geq G^{(1)} \geq G^{(2)} \geq G^{(3)} \geq \dots \...
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1answer
71 views

Surjective mapping of matrices under rotational and reflection symmetries

Let me preface this by saying that I'm not a mathematician and that I'm having a hard time stating my problem in the proper terms. Nevertheless, I'm faced with a problem for which I think an elegant ...
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18 views

Is $\prod^t_i S_{N_i} \wr D_{m_i}$ a normal subgroup of $S_{\sum^t_i N_i m_i}$?

How can I determine whether $\prod^t_i S_{N_i} \wr D_{m_i}$ a normal subgroup of $S_{\sum^t_i N_i m_i}$? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols,...
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How to determine the order of $\sum^t_i S_{N_i} \wr D_{m_i}$?

How can I determine the order of $\sum^t_i S_{N_i} \wr D_{m_i}$ ? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral group of ...
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16 views

How to prove $\sum^t_i S_{N_i} \wr D_{m_i}$ non-Abelian?

How do I prove that $\sum^t_i S_{N_i} \wr D_{m_i}$ is a non-Abelian group? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral ...
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78 views

Exponent of the direct sum of finite groups, specifically, $\sum^t_i S_{N_i} \wr D_{m_i}$

I have one general and one specific questions. What is the expression for the exponent of the direct sum of finite groups? What is the exponent of $\sum^t_i S_{N_i} \wr D_{m_i}$? Here, $i, N_i, m_i$ ...
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27 views

Relation between direct product and direct sum of $S_{N_1} \wr D_{m_1}, \ldots, S_{N_t} \wr D_{m_t}$

I am trying to understand the relation between the direct sum and direct product of all the groups from the set $$\{S_{N_1} \wr D_{m_1}, \ldots, S_{N_i} \wr D_{m_i}, \ldots, S_{N_t} \wr D_{m_t}\}$$ ...
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626 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. $\mathbb{Z}_2 \cong \{e,j\} \subset D_n$ ...
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44 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 ...
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1answer
74 views

Simple homomorphism Question: Prove that $φ(s)$ is a reflection

Let r ∈ D20 be an element of order 20 and let s ∈ D20 be a reflection. Suppose that φ : D20 → D20 is a homomorphism such that φ(r) = r^12 Prove that $φ(s)$ is a reflection
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1answer
236 views

Dihedral group of order 2n

I would appreciate if someone could prove this for me: Let G be a dihedral group of order 2n and suppose H is a cyclic quotient group of G. Show that |H|is less than or equal 2.
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1answer
53 views

Find all $x$ in $D_4$ such that $ax(a^{-1}) = b$.

Consider the dihedral group $D_4$. Consider also the elements $a= r_1$ and $b= S_1$ of $D_4$. Find all $x$ in $D_4$ such that $ax(a^{-1}) = b$. Do both $a$ and ($a^{-1}$) cancel each other out? If not,...
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70 views

Is every symmetric group generated by some cycles?

Actually, I have two questions here: How to show that $D_3$, the dihedral group, is isomorphic to $S_3$ in details? Is every symmetric group generated by some cycles? For question (1), I know that, ...
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1answer
116 views

The Dihedral group $D_1$ is non-abelian?

Same as above. I'm trying to show that for any n being odd, $D_n$ has exactly n elements of order 2 where $D_n$ is non-abelian. I know that for $n\ge3$ this is true, but what about for $n=1$.
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63 views

Be $G$ a group of order $|G|=2p$ where $p$ is a prime number odd, prove that either $ G $ is cyclic or dihedral $G\simeq D_p$ the group of order 2$p$. [duplicate]

Be $G$ a group of order $|G|=2p$ where $p$ is a prime number odd, prove that either $ G $ is cyclic or dihedral $G\simeq D_p$ the group of order 2$p$. I thought the question a little difficult and ...
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1answer
185 views

What is the formula for products in dihedral groups?

Let $G \colon = \langle x, y \ | \ x^2 = y^n = e, \ x^iy^j = x^{i^{\prime}} y^{j^{\prime}} \ $ if and only if $ i = i^{\prime}, j = j^{\prime}, \ xy = y^{-1}x \rangle$. That is, let $G$ be the ...
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128 views

Lines of reflection in a Dihedral group, Why is this paradox happening?

So, we know that in a Dihedral Group $D_n$, if $r$ represents counterclockwise rotation by $2\pi/n$ radians and $s$ is any axis of reflection, then the elements of reflection stand as follows: {$s, ...
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1answer
27 views

Why isn't $r^{\frac{n}{2}}$ classified as an involution?

Suppose you have a $(2n)$-gon for some $n \in \mathbb{N} : n > 1$. Then the rotation $r^{\frac{n}{2}}$ where $n$ is the number of vertices imposed on the $(2n)$-gon is the same as an involution. ...
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1answer
109 views

Question about definition of generator for dihedral groups?

Dummit and Foote mention that a relation for the dihedral group is $rs = sr^{-1}$. Now, I have interpreted the statement to mean $r$ is a rotation of $\frac{2\pi}{n}$ radians and $s$ is an involution....
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1answer
82 views

I'm trying to understand the following part from Gallian text

I'm trying to understand the following part (Chap. Sylow Theorem, Paragraphs preceding the article Application of Sylow Theorem) from Gallian text I'm trying to understand the why. That is I need to ...
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2answers
211 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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1answer
50 views

Let $D_{2n}$ be the dihedral group of order $2n$. Let $H$ be the set of rotations of the regular $n$-gon. Is $H\lhd D_{2n}$? [closed]

Let $D_{2n}$ be the dihedral group of order $2n$, i.e., the group of symmetries of the regular $n$-gon. Let $H$ be the set of rotations of the regular $n$-gon. Is $H\lhd D_{2n}$?
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2answers
36 views

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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1answer
40 views

Group G with elements of specific order [closed]

What is an example of a group G with elements a and b such that the order or a and b is 2, but the order of ab is 3? I'm thinking some sort of Dihedral group perhaps?
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1answer
36 views

Prove the existence of homomorphism.

I am trying to answer the following question. Is there any group homomorphsim $\phi: D_4 \rightarrow S_5$?
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1answer
52 views

Is dihedral group defined as following commutative?

Let $G$ be the dihedral group defined as the set of all formal symbols $x^iy^j$, $i=0,1$, $j=0,1,\ldots,n-1$, where $x^2=e$, $y^n=e$, $xy=y^{-1}x$. EDIT - My proof is wrong .But i will be thankful to ...
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1answer
310 views

Prove that the group D12 has no element of order 12 [closed]

Is there a way to prove this without checking each element of the group one by one?
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1answer
841 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 ...