# 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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### 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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### 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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### 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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### 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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### 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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### $D_{2n}$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...