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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Proper condition on the dihedral group

Is there a theream which is a condition on $n\in\mathbb N$ that says when the dihedral group, $D_{n}$, has non-cyclic subgroups? After spending some time figuring a condition I tried to find some ...
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169 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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1answer
313 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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3k views

Subgroups of $D_3$

What are all the subgroups of $D_3$, I have so far if $D_3=\{r_0, r_1,r_2, s_1, s_2, s_3\} $ then the subgroups are $$\{r_0, r_1, r_2\},\{r_0, s_1\}, \{r_0, s_2\}, \{r_0, s_3\}, D_3, \{r_0\}. $$ Are ...
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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 ...
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822 views

How many proper nontrivial subgroups do D5 have?

Do I have to find out every element in D5 and draw a table to find out subgroups? I know how to find out every single element in D5, but can't think of how to find proper nontrivial subgroups
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167 views

let $G$ be dihedral group so that $D_{2n}= \langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. Prove that $N\unlhd G$ and $G/N \cong W = \{1,-1\}$

let $G$ be dihedral group so that $G = D_{2n}=\langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. 1) let $N < G$ so that $N =\langle y \rangle = \{e,y,...y^{n-1}\}$. Prove that $G \unlhd N$. 2) $...
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412 views

Alternative way of proving the subgroup of rotations is normal in $\mathbb D_4$

I've just solved a basic group theory exercise which is: decide if $\{1,r,r^2,r^3\}$ is a normal subgroup of $\mathbb D_4$ (I mean the dihedral group of $8$ elements, not the one of $4$). I've used ...
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1answer
29 views

Orders of the Elements of $D_6/Z(D_6)$

I have been trying to calculate the orders of the elements of $D_6/Z(D_6)$. For example, using $R_{60}$ to represent rotation by 60 degrees and $R_0$ to represent rotation by 0 degrees (the identity ...
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71 views

Quotient group of dihedral group

Let $G=\{e,r^{2},...,r^{8},s,sr,...,sr^{8}\}$ and let $N=\langle r^{3} \rangle.$ Now let $\pi(g)=\bar{g}=gN$ be surjective with kernel $N$. I have to show that $G/N=\{\bar{e},\bar{r},\bar{r^{2}},\...
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38 views

Compute a set $S$ given information about how it is acted upon transitively by $D_8$

Let $D_8=D_{2 \cdot 4}$ be the dihedral group on a regular $4$-gon. Suppose that $S$ is a subset of $S_4$, such that S contains the element $( 1 \ 2 \ 3)$. We also know that $D_8$ acts transitively ...
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38 views

A formula for the number of order $2$ elements of $D_m\times D_n$ for even $m>2$ and odd $n>2$. (Gallian 8.24.)

This is Exercise 8.24 of Gallian's "Contemporary Abstract Algebra (Eighth Edition)". Answers that use material from the textbook prior to the exercise are preferred. Presentations, for instance, are ...
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89 views

Quotient group of the dihedral group by $\langle r^2 \rangle.$

Show that $G/H$ is abelian, where $G$ is the dihedral group $$ G={\langle r,\, f \mid r^n=f^2=1,\, rf=fr^{-1}\rangle}$$ and $H$ is the subgroup $\langle r^2 \rangle.$ I've tried showing that for $...
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86 views

Find a group $G$ with $a\in G$ such that $|a|=6$ but $C_G(a)\neq C_G(a^3)$.

This is part of Exercise 46 of Chapter 3 of Gallian's "Contemporary Abstract Algebra". Notation 1: The centraliser of $g$ in a group $G$ is denoted $C_G(g)$. Notation 2: The dihedral group $...
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214 views

Is it only the generator of the group that commutes with all the other elements?

If a group is generated by an element does that mean the generator commutes with all the other elements or does it mean that because the group is cyclic(as it has a generator) that all elements ...
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126 views

Show that a dihedral group of order $4$ is isomorphic to $V$, the $4$ group. [closed]

Show that a dihedral group of order $4$ is isomorphic to $V$, the $4$ group. Also, please show that a dihedral group of order $6$ is isomorphic to $S_3$. Thank you!
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125 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
90 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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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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112 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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125 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 $...
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236 views

How many homomorphisms are there from $D_6$ to $D_5$?

I know that: $D_6$={$e,a,a^{2},a^{3},a^{4},a^{5},b,ab,a^{2}b,a^{3}b,a^{4}b,a^{5}b$}, with $a^{6}=e$ and $ba^{k}b=a^{-k}$. $D_5$={$e,r,r^{2},r^{3},r^{4},s,rs,r^{2}s,r^{3}s,r^{4}s$}, with $r^{5}=e$ ...
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191 views

Find the length and composition factors of $D_n$

Let $D_n$ denote be the dihedral group with $2n$ elements. Let $n = p_1^{e_1}\cdots p_r^{e_r}$. I need to prove that $$l(D_n) = e_1 + \cdots +e_r +1$$ and $$\mathrm{fact}(D_n) = \{C_{p_1},...(e_1\text{...
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232 views

How many subgroups of order $n$ does $D_n$ have?

How many subgroups of order $n$ does $D_n$ have? My work: Since subgroups of $D_n$ are either cyclic or dihedral, the subgroups of order $n$ of $D_n$ are $\left< r \right>$ (cyclic) and $D_{n/...
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136 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 (...
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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 ...
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3k views

How many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads?

I am trying to figure out how many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads. I understand how to do the problem with two colors, but I am struggling to ...
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478 views

Automorphisms in the Dihedral groups

Let g be a group and $a \in G$. Define $\phi_a:G\rightarrow G$ by $\phi_a(g)=aga^{-1}.$ Now Let $G=D_4$ and $a=r$, where $r$is the rotation. We must show that $\phi_r: D_4\rightarrow D_4$. So show ...
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589 views

Given a Cayley table, is there an algorithm to determine if it is a dihedral group?

Showing that it is a group is simple enough, but is it possible to determine if it is a dihedral group or not just by looking at the Cayley table?
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4k views

Isomorphism between dihedral group and a subgroup of $S_n$

I need to find an isomorphism between $D_n$ (all symmetries of an $n-gon$) and a subgroup of $S_n$. I know that Cayley's theorem gives a nice isomorphism that shows that $D_n$ is isomorphic to a ...
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1answer
1k views

Dihedral group as a direct product

In a problem from a past exam I am asked "When can $D_n = \langle r,s\mid r^n = s^2 = (rs)^2 = 1\rangle$, the dihedral group of order $2n$, be expressed as a direct product $G\times H$ of two ...
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78 views

Find the inner and outermorphisms of a particular dihedral group

Given that |Inn($D_8$)| = 8 and |Out($D_8$)| = 2 where Out($D_8$) = Aut($D_8$)/Inn($D_8$) and $D_8$ = {e,r,$r^2$,..,$r^7$,s,sr,...,$sr^7$} we want to find Inn($D_8$) and Out($D_8$). We know that Out(...
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50 views

How many homomorphisms are there from $D_5$ to $V_4$?

Question: How many homomorphisms are there from $D_5$ to $V_4$, where $D_5$ is the dihedral group of order $10$ and $V_4$ the Klein four-group? I've used the fact that since $V_4$ is abelian, the ...
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1answer
22 views

For all $n > 0$ express $D_{2n}$ as a semidirect product $\mathbb Z_n \rtimes_\theta \mathbb Z_2$, finding $\theta$ explicitly.

For all $n > 0$ express $D_{2n}$ as a semidirect product $\mathbb Z_n \rtimes_\theta \mathbb Z_2$, finding $\theta$ explicitly. I am not sure how to go about finding $\theta: \mathbb Z_2 \to \...
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35 views

Show the Dihedral Group $D_n$ is generated by rotations and reflection along the x axis.

I'm having problems understanding the excersice: E) Define $D_n$ as the group of symmetries of a regular n-gon. Name the vertices $V=\{V_0,V_1,...,V_{n-1}\}$ so that $$V_{k}=\exp({i\cdot\dfrac{2\pi k}{...
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26 views

Set of Rotations Cyclic?

For the dihedral group $D_{n}$ of order $2n$, is the group $R$ formed by its $n$ rotations cyclic in general? Or is the factor group $D_{n}/R$ cyclic? I am trying to show the series $D_{n}>R>(1)$...
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58 views

Coloring sides of truncated triangular dihedral(bipiramid) into 3 colours

I need to find out the amount of ways to colour truncated triangular dihedron into 3 colours. So, the task will be easier if I had simple triangular dihedron. First of all, do I understand right ...
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72 views

Finding Homomorphisms from dihedral groups to cyclical groups

Ok so there was another question very similar to this on here however it leaves me a little confused. $\bf{Question}$ Let G = $D_{14}$ the Dihedral group order 14 and A = $c_7$ be the cyclical ...
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1answer
27 views

Normal subgroup generated by $D_8$

Let $D_8 = <a,b : a^4 = b^2 = 1, bab^{-1} = a^{-1}>$ be the dihedral group. I'm trying to show that the subgroup generated by $a^2$ is normal. But, isn't $<a^2> ={\{1, a^2}\}$? So the ...
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1answer
22 views

Counting the elements of a dihedral group $D_n$ of a $n$-sided regular polygon

For a $n$-sided regular polygon there are $n-1$ possible rotations: $a,a^2,a^3,a^{n-1}$, a 1 reflection $b$, 1 identity $e=a^n=b^2$. There are also 2(n-1) elements $ab,a^2b,a^3b,...a^{n-1}b$ and $ba,...
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64 views

In dihedral group $FR=R^{-1}F$

Let F be any reflection (flip ) about axis of symmetry . And R be rotation by $\frac{2\pi}{n} $ radian counterclockwise .(n is the number of vertex). Then $FR=R^{-1}F$ I looked at some example and ...
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56 views

conceptual question with example: proving groups are isomorphic

The question is to prove $D_8$ and the subgroup of $S_4$ generated by $(1 2)$ and $(1 3)(2 4)$ are isomorphic. I was able to show that the relations for $D_{8}$ follow when we set $b = (1 2)(1 3)(2 4)...
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76 views

Color Vertices of a Regular Pentagon with Two Colors

I'm trying to figure out how many distinct ways there are to color the five vertices of a pentagon two different colors. I know this requires the use of Burnside's theorem, but am struggling a bit ...
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1answer
82 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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201 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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1answer
304 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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495 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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349 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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93 views

How to geometrically show that there are $3$ $D_4$ subgroups in $S_4$?

As shown in this note, the symmetry group $S_4$ for a cube has $3$ subgroups that are isomorphic to $D_4$, the dihedral group of order $2 \times 4 = 8$. How to geometrically illustrate this fact? ...
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153 views

Expressing the action of the dihedral group D4's generators as matrices, with respect to a certain basis.

$D_4$ is generated by a rotation $\alpha$ of order 4 and a reflection $\beta$. Its elements $e$, $\alpha$, $\alpha^2$, $\alpha^3$, $\beta$, $\alpha\beta$, $\alpha^2\beta$, $\alpha^3\beta$ give an ...