# Tagged Questions

For questions on dihedral groups, the group of symmetries of a regular polygon, including both rotations and reflections

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### Automorphism groups of partially cycle graphs

I define partially cycle graphs as follows. If we add the same subgraph to $n-k$ vertices of an $n$-vertex cycle graph, where $1\le k < n$, we create a partially cycle graph. Here are a few ...
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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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### Number of orbits with Burnside's lemma

We color a equilateral triangle by coloring each edge with one of $k \geq 1$ colors. Find a formula for the number of orbits under the action of $D_6$, the dihedral group of $6$ elements, on the ...
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### Can the dihedral groups be seen as subgroups of each other?

I am solving one difficult problem and now I need information on this: If $m\mid n$, is then possible to "embed" $D_m$ to $D_n$, or otherwise said, does $D_n$ have a subgroup isomorphic to $D_m$? ...
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### Non-trivial graph automorphism groups with $D_n$ as subgroup

I understand that the automorphism group of an $n$ cycle graph is the dihedral group $D_n$ of order $2 n$. From the comment of @Christian, I also understand that $S_n$ is the automorphism group of the ...
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### Dihedral groups: Relationship between symmetries and rigid motions

In Dummit & Foote, $D_{2n},\ n \geqslant 3$ is the set of symmetries of a regular $n$-gon, where a symmetry is a rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-gon, ...
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### How can we find all the subgroups?

I want to find all the normal subgroups of $D_n$. We have that $K$ is a normal subgroup of $D_n$ iff $$gkg^{-1}=k\in K, \forall g\in D_n \text{ and } \forall k\in K$$ right? Could you give me some ...
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### Classification of the irreducible group representations of the dihedral groups

Let $D_n$ be the dihedral group of order $2n$. Show that all irreducible representations have vector space dimension $1$ or $2$, and describe them up to isomorphism. Any hints how to even start?
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### Order of a certain finitely generated group

Suppose I am looking at the group $W=\langle s_\alpha, t_\beta \rangle$ where $s_\alpha$ and $t_\beta$ are reflections in $\mathbb{R}^2$ coming from two vectors $\alpha$ and $\beta$ making an angle of ...
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### Determine all the subgroups of the dihedral group $D_{15}$

Is there an algorithm for finding all of the subgroups of $D_{15}$? Also, is there a formula for finding the size of that subgroup? Not sure where to start with finding all the subgroups of $D_{15}$ ...
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### normaliser of dihedral group of order 4

Let $D_{2n}$ be the dihedral group of order $2n$. I would like to show that if $4$ divides $n$, then the normaliser $N_{D_{2n}}(D_{4})=D_{8}$. I know that $$|D_{2n}:N_{D_{2n}}(D_{4})|=|\mathcal{C}|,$$...
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### Multiplication table of $(S_1 \wr D_2) \times (S_1 \wr D_3)$

I am trying to compute the multiplication table of $(S_1 \wr D_2) \times (S_1 \wr D_3)$. My effort: I understand that $S_1$ is the trivial group consisting only the unit element i.e. $e$. $e$ will ...
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### Are subgroups of automorphism groups of trees direct product of symmetric, cyclic and dihedral groups?

My question is triggered by my confusion with the notation $\Psi$ in Constructive Approach to Automorphism Groups of Planar Graphs by Klavík et al. The notation $\Psi$ was first used expressing ...
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### Relation between $\prod^t_i S_{N_i} \wr D_{m_i}$ and $A_{\sum^t_i N_i m_i}$?

$\prod^t_i S_{N_i} \wr D_{m_i}$ is a subgroup of the symmetric group $S_{\sum^t_i N_i m_i}$. Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ ...
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### Can there be a tower like $1 = A_0 \triangleleft \ldots \prod^t_i S_{N_i} \wr D_{m_i} \ldots \triangleleft = S_{\sum^t_i N_i m_i}$?

$\prod^t_i S_{N_i} \wr D_{m_i}$ is a 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, $D_{m_i}$ is the dihedral ...
### Is $\prod^t_i S_{N_i} \wr D_{m_i}$ a maximal subgroup of $S_{\sum^t_i N_i m_i}$?
How can I determine whether $\prod^t_i S_{N_i} \wr D_{m_i}$ is a maximal 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 $N_i$ ...