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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42 views

Finite group $G$ with $\exp(G)=2^{n-1}$

Let $G$ be a finite non abelian group of order $2^n$ and exponent $2^{n-1}$. What can we say about $G$ ? Does $G$ isomorphic either to the Dihedral group $D_{2^n}$ or to the generalized Quaternion ...
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1answer
55 views

Compute the order of each of the elements in $D_6$ where $D_{6}=\left\langle r, s \mid r^{3}=s^{2}=1, r s=s r^{-1}\right\rangle$

Compute the order of each of the elements in $D_6$ where $D_{6}=\left\langle r, s \mid r^{3}=s^{2}=1, r s=s r^{-1}\right\rangle$ I found six elements of $D_6$ are $1,r, r^2,s, rs, r^2s.$ How can I ...
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1answer
55 views

Is $\operatorname{Aut}(D_{12})\simeq D_{12}$?

Let $D_{12}$ be the dihedral group of order 12. Then $$|\operatorname{Aut}(D_{12})|=6\phi(6)=12=|D_{12}|,$$ and the standard method of proof for $$\operatorname{Aut}(D_6)\simeq D_{6}\qquad\mbox{and}\...
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3answers
40 views

Describing homomorphisms from $\Bbb Z_n$ to $D_m$.

I've been asked to find all group homomorphisms from $$\Bbb Z_n\to D_m,$$ where $n$ and $m$ are distinct natural numbers. I now understand how to describe homomorphic groups using functions between ...
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0answers
21 views

More detail proof of unique factorization of dihedral group.

Prove that every element in $D_{2n}$ has a unique factorization of the form $a^{i}b^{j}$, where $0 \leq i < n$ and $j=0$ or $1.$ From Unique factorization of dihedral group, a answer has mentioned ...
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0answers
26 views

Using an operations table for the group $D_{6}$

With $H = \{ \rho_{0} , \rho_{3} \}$, how would I compute $( \mu_{1} , H)(\mu_{2} , H)$ and $( \mu_{2} , H)(\mu_{1} , H)$? I'm assuming I'd calculate $( \mu_{1} , \rho_{0} )( \mu_{2} , \rho_{3} )$ and ...
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1answer
109 views

Number of subgroups isomorphic to $\mathbb{Z}_{2}\times \mathbb{Z}_{2}$ in $D_8$.

Let $$D_8=\langle \sigma,\rho \; | \; \rho^8=\sigma^2=e \text{ and } \sigma\rho\sigma=\rho^{-1} \rangle $$ be the dihedral group of 16 elements. First of all I found the number of elements of order 2 ...
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2answers
81 views

how to verify $fh=h^{-1}f$ for dihedral group of order $2n$

Let $S$ be the plane, that is, $S=\{(x,y)|x,y∈R\}$ and consider $f,h\in A(S)$ defined by $f(x,y)=(−x,y)$. Let $n>2$ and let $h$ be the rotation of the plane about the origin through an angle of $\...
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1answer
30 views

find a formula for $(f^ig^j)∗(f^sg^t)=f^ag^b$ that expresses $a,b$ in terms of $i,j,s$ and $t$

Let $S$ be the plane, that is, $S=\{(x,y)|x,y∈R\}$ and consider $f,g∈A(S)$ defined by $f(x,y)=(−x,y)$ and $g(x,y)=(−y,x)$; $f$ is the reflection about the $y−$axis and $g$ is the rotation through $90^{...
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0answers
30 views

Order of elements of the dihedral group

Let $G= ${$e, r, r^2, . . . , r^{23}, s, sr, sr^2, . . . , sr^{23} $} be the dihedral group with $48$ elements. (a) Compute which elements of $G$ have order $3$. (b) Determine if $H =$ {$r^i$ with $i$...
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43 views

Prove the reflections of the dihedral group can be written in the form $r^hs$

I saw in some exercise that the 2n elements of the dihedral group $D_n$ were written as $1 , r, r^2, ... , r^{n−1}, s, r s, r^2s, ... , r^{n−1}s$, $r$ being the $2\pi /n$ counter-clockwise rotation ...
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1answer
27 views

Are symmetry axes in $D_n$ fixed?

I am studying the dihedral group and comparing the table I have in my book with that of wikipedia. I realize that the entries were inverted, that is XY in the wikipedia article corrisponds to YX in my ...
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2answers
92 views

Show that the dihedral group $ D_3 $ has only one subgroup of order 3

The extra hint was that the order $ \circ (g) $ of $g$ divides the order of $ \circ (G) $ of $G$ I claimed that the subgroup of order $3$ in $G$ was $\langle\,a\,\rangle_{a^3=e}$ which was obvious. ...
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1answer
29 views

Prove that if there is a nontrivial homomorphism from $D_n$ to G then the order of G is even.

Let $D_n$ denote the dihedral group of order 2n. Let G be a finite group. Prove that if there is a nontrivial homomorphism from $D_n$ to G then the order of G is even. I am thinking to use $D_n/ker(\...
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0answers
44 views

Dihedral group as a semidirect product?

It is known that the dihedral group $D_{2n}$ is isomorphic to the semidirect product $Z_n\rtimes Z_2$, where both $Z_n,Z_2$ are cyclic. My question is, for a semidirect prouct, the two subgroups ...
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1answer
47 views

Action of the symmetry group on polygons

I want to show that a polygon $P$ is regular, i.e. all the angles are equal, iff $\Sigma(P)$, the symmetry group, acts transitively on Vert$(P)$.
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31 views

How to identify the two copies of $D_{24}$ in the homomorphisms of the 2 musical actions?

Let $S$ be the set of minor and major triads. Two sets of actions are defined on the set: 1) Musical transposition and inversion 2) P, L, R actions $P(C-major) = c-minor,$ $L(C-major) = e-minor,$ ...
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1answer
40 views

Transforming a permutation into a rotation

Let $n\geq 3$ and $G=C_{n}=\{1,r,...,r^{n-1}\}$ be the cyclic group of $n$ elements where $r$ is the rotation of $360/n$ degrees. Here, let us consider a vector $x\in\mathbb{R}^{n}$ as consisting of ...
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46 views

What are some nice informations about the dihedral groups,alternating groups,symmetric groups.

I am an undergraduate student and I want to know some nice informations about some special groups like the dihedral group $D_{2n}$,of regular $n$-gon , alternating group $A_n$ and symmetric group $S_n$...
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64 views

Example of dihedral groups with same order

I have to prove or give counter example "Is it true that if $|G_1|$ and $|G_2|$ are dihedral groups of order $|G_1|=|G_2|$ then G1≅G2 " $D_{2n}=<a,b\quad | \quad a^n=b^2=1 \quad ba=a^{-1}b>$ ...
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2answers
33 views

Using Definition of Cyclic Group to prove B is a Subgroup

Given the Dihedral group $ D_4 $ (that is where $ D_4 = $ { $ id, R, R^{2}, R^{3}, F, RF, R^{2}F, R^{3}F $} ); Let $B =$ {$id, RF$} I now wish to prove that $B$ is a subgroup of $D_4$: Note that $B =...
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1answer
83 views

Solving Functional Equation Involving Substitution

Suppose we seek a function of two variables $f(x,y)$ such that $f(x,y) = f(y,x)$ $f(x,y) = f\left(\frac{1+y}{x}, ~ y \right)$ Are there known techniques for approaching such questions? I already ...
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1answer
37 views

Binomial theorem and multinomial coefficient

I have a question to which I could not find an answer in the forum. I am trying to solve a bracelet problem. Actually I already solved it but with some kind of a cheating (using wolframaplha to ...
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1answer
19 views

Compound angles formula derivation(crown molding)

So I've been trying to get my head around this for a week now. It's a practical problem, but the geometry seems more involved then I initially thought. When you want to attach a crown molding to a ...
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2answers
41 views

Help Understanding Why $D_{10} = C_2$ $\times$ $D_5$ [duplicate]

I was running around the internet looking at Dihedral groups and ran into a page that claimed that $D_{10} = C_2$ $\times$ $D_5$ (I know notation for Dihedral groups is confusing - here I'm using $D_{...
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0answers
57 views

Let $G$ be the set of the 6 axial symmetries and the 6 axial rotations of an hexagon $E$ that transform $E$ into itself

Let $E$ be a hexagon whose side's measure is $1$ and let $T$ be the set of all the triangles whose vertices are three of the vertices of the hexagon. Then, let $G$ be the set of the six axial ...
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1answer
32 views

Dihedral subgroup of a infinite Coxeter group

I have seen a conclusion that every infinite Coxeter group contain an infinite dihedral subgroup, but I have no idea how to prove it. Could anyone give me some hint?
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1answer
73 views

Induced Group Representation

Let $D_{\infty} = \langle a,t \mid a^2 = t^2 = 1 \rangle$ be the infinite dihedral group, and let $H = \langle at \rangle$. Given $\theta \in [0,2 \pi)$, let $f_{\theta} : H \to \Bbb{T}$ be defined ...
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2answers
82 views

Prove that $\mathbb {D}_4 $ is isomorphic with $\mathbb {Z}_2 \times\mathbb{Z}_2 \times\mathbb{Z}_2$ [closed]

I tried comparing characteristics between the Dihedral group and the $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ group, but I need help to be able to define the function and be able to ...
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1answer
76 views

Symmetries group of hexagon

i'm trying to understand how to build a group H that contains complex functions and its operation is function composition. I really don´t undestand how to build that group for a hexagon in the ...
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2answers
34 views

What are the elements of dihedral group $\mathbb{D_6}$? [duplicate]

I'm not really all that good with dihedral groups and can't find anything online that explicitly shows the elements of $\mathbb{D_6}$
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4answers
68 views

Can a group have more than one operation? If not, what is the operation of a dihedral group?

In my 1st year Mathematics BSc course, dihedral groups seems to include rotations and reflections, which suggests that groups can have more than one operation. But definitions of groups I've seen seem ...
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1answer
62 views

Does $\{a,b,c,c^2\}$ generate the same group as $\{a,b,c\}$?

Is generated group by $\{a,b,c,c^2\}$ same as group generated by $\{a,b,c\}$? I think the answer is YES. But here is a paragraph of J. Wolf's Book: Let $\triangle_8$ denote the regular octahedron (...
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0answers
7 views

Form of elements in dihedral group.

Show that any element in Dihedral group $D_{2n}=\langle x,y:x^n=y^2=xyxy=e\rangle$ is of the form $y^{a}x^{b}$ with $a\in\left\{0,1\right\}$ and $b\in \left\{1,\ldots,n\right\}$. I have this: ...
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0answers
16 views

Any rotation in dihedral group is of the form $\left\{x,x^2,\ldots, x^{n-1},e\right\}$

Let $D_{2n}, n\geq 3$, be the dihedral group with $2n$ elements. It is generated by $x, y$, satisfying $x^n = y^2 = xyxy = 1$. Prove (algebraically) that every element not in the subgroup $\langle x\...
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1answer
40 views

Choice of generator in dihedral group

I'm about to begin studying group representation theory, and I want to get more familiar with the symmetric group $\mathfrak{S}_n$ (and its subgroups) first. In particular, I'd like to better ...
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1answer
127 views

Branch points of a dihedral Galois branched cover of a complex torus

Let $\Lambda$ be a lattice in $\mathbb{C}$ and $X = \mathbb{C}/\Lambda$ be a complex torus. Exercise 6 of chapter 3 of Tamás Szamuely's book "Galois Groups and Fundamental Groups" (actually, the ...
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22 views

Invariant polynomials under dihedral group action

I'm trying to solve the following problem: Find a generating set for the algebra of invariant polynomials $\mathbb C[x_1, x_2]^\Gamma$, where $\Gamma$ is a dihedral group $D_n$, generated by ...
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1answer
52 views

Geometric interpretation of conjugacy classes and class equation of $D_6$ [closed]

Known that for Dihedral Group $D_6$, where $D_6=\{r,s: r^6=s^2=1, rs=sr^{-1}\}$, its conjugacy classes are given by $\{1\}, \{r,r^5\}, \{r^2,r^4\}, \{r^3\}, \{s, sr^2, sr^4\}, \{sr, sr^3, sr^5\}$, ...
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0answers
29 views

Use direct product theorem to prove $D_{12}$ is isomorphic to $D_6 \times C_2$ [duplicate]

Show that the dihedral group $D_{12}$ is isomorphic to $D_6 \times C_2$ I have solved this question by using Direct product theorem, but there is a question. When I express $D_{12}$ as its usual way,...
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3answers
48 views

What exactly is the only Sylow $3$-Subgroup of $D_3$?

I can apply the needed theorem to get me to the fact that it only has one Sylow $3$-Subgroup but I donʻt know how to find exactly what it is. I have the multiplication table computed so help in ...
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2answers
101 views

Are the Dihedral Groups of order $n!$ isomorphic to $S_n$?

It is well known that the symmetries of a triangle, which is the Dihedral Group of order 6, is isomorphic to $S_3$. It is clear that both of these have 6 elements. However, $D_4$, the symmetries of ...
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2answers
84 views

Using Group Actions to determine the different colourings of a grid

I am trying to find the number of different colourings of an $n \times n$ grid using $m$ colours by using group actions. The set that I am acting on is the set of $n^2$ ‘tiles’ within the square grid. ...
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2answers
90 views

If $n \geq 3$ then there is no surjective homomorphism $f: D_{2n} \to Z_n$.

Claim: If $n \geq 3$ then there is no surjective homomorphism $f: D_{2n} \to Z_n$. In this case $D_{2n}$ refers to the dihedral group of order 2n. Thoughts: I'm thinking that the proof to this ...
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2answers
38 views

Understanding the Dihedral group, $D_n$

I know what the dihedral group is, the group of symmetries (length-preserving) functions of a regular n-gon. However, when it is referring to the symmetries of the regular n-gon what is it referring ...
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22 views

Isomorphic subgroup of 2n-gon

Consider a regular 2$n$-gon $P_{2n}$ with vertices denoted $a_1, a_2,...,a_n$, by going in the clockwise orientation. Consider the subset of vertices of $P_{2n}$ given by $S$ = {$a_2,a_4,a_6,...a_{2n-...
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1answer
46 views

Surjective homomorphism from the infinite dihedral group to each finite dihedral group

The infinite dihedral group $D_\infty$ is a subgroup of permutations of the integers generated by $f(n) = -n$ and $g(n) = 1-n$, which reflect the integer number line over the point 0 and 1/2 ...
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0answers
41 views

Prove that $D_{2n}$ is a subgroup of $S_n$ [duplicate]

I have the idea that if I define $r=(a_1,a_2,...,a_n)$ and $s=(a_i,a_j)$ then I am probably done . How do I proceed from here ? And how do I write the proof properly?
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1answer
36 views

Proving that subgroup of Rotations in $D_{2n}$ is a normal subgroup [duplicate]

I am trying to prove that the group of rotations in $D_{2n}$ is a normal subgroup. I know this group has a cardinality $n$, and the ratio of the $D_{2n}$ and $R$ is $2.$ I am trying to construct a ...
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0answers
53 views

Action of the dihedral group $D_{2n}$ on vertex set of regular $n$-gon is primitive $\iff\,n$ is prime?

I'm attempting to prove that the action of the dihedral group, $D_{2n}$, is primitive on the set of vertices of the regular $n$-gon $\iff\,n$ is prime. I'm not wholly sure where to start. I know that ...

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