# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...