Questions tagged [dihedral-groups]

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

383 questions
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What is the required group theory knowledge needed to understand Verhoeff's algorithm?

The Wikipedia page tells me I need to understand permutation groups and dihedral groups. Can someone clearly outline what exactly the perquisites of understanding this is and how much time I'll take ...
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All groups of order 12; Dic12 and D6

I know this has been asked in various forms before, but so far I have failed to understand those answers properly. I've also read several papers discussing this, but I don't really get it. I have an ...
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Finding normal subgroups

Let $G=\langle e, r,..., r^{n-1},s,sr,...,sr^{n-1} \rangle$ be a dihedral group with $2n$ elements, for $3 \leq n$. Prove that the only normal subgroups of $G$ are $\langle r^d \rangle$ (where $d$ ...
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Row in the character table of $D_{10}$

Give the values of one row of the character table of $D_{10}$ corresponding to a character of degree $2$ I know the conjugacy classes of $D_{10}$, the dimensions of the irreducible representations ...
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Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
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Surjective mapping of matrices under rotational and reflection symmetries

Let me preface this by saying that I'm not a mathematician and that I'm having a hard time stating my problem in the proper terms. Nevertheless, I'm faced with a problem for which I think an elegant ...
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Normal subgroup test

Hi there I have this problem: Is $<p^6\epsilon^5>$ a normal subgroup of the Dihedral group $D_4 = \{ I,p,p^2,p^3,\epsilon, p\epsilon, p^2\epsilon,p^3\epsilon \}$? Since I'm not that good at ...
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Dihedral groups acting on Riemann surfaces

I'm studying the quotient riemann surface $X/G$. I'm looking for examples of dihedral groups $D_n$ acting on some riemann surfaces $X$ or at least acting on it's Jacobian JX. Does anybody knows some ...
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Prove a group that has a normal subgroup isomorphic to $D_8$ has a non-trivial center

Let $G$ be a group which has a normal subgroup isomorphic to $D_8$. Prove that $G$ has a non trivial center. So, given $g\in G$, $h\in D_8$ $ghg^{-1}\in D_8$. So I tried to prove that there is an ...
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How are the elements of a dihedral group usually defined?

While searching online, I've come across two ways to define the elements of the dihedral group. Both ways are internally consistent and are fine as far as I can tell, but they are mutually exclusive, ...
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Dihedral groups are solvable [duplicate]

I'm trying to prove the dihedral groups are solvable for any Dn. I use the normal subgroup of all rotations, since the quotient of Dn/{rotations} is isomorphic to Z2 so it's abelian as well, so we get ...
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Exponent of the direct sum of finite groups, specifically, $\sum^t_i S_{N_i} \wr D_{m_i}$

I have one general and one specific questions. What is the expression for the exponent of the direct sum of finite groups? What is the exponent of $\sum^t_i S_{N_i} \wr D_{m_i}$? Here, $i, N_i, m_i$ ...
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Relation between direct product and direct sum of $S_{N_1} \wr D_{m_1}, \ldots, S_{N_t} \wr D_{m_t}$

I am trying to understand the relation between the direct sum and direct product of all the groups from the set $$\{S_{N_1} \wr D_{m_1}, \ldots, S_{N_i} \wr D_{m_i}, \ldots, S_{N_t} \wr D_{m_t}\}$$ ...
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What can we say about $D_{2n}$ and $D_n$ if $n$ is even?

We know that if $n$ is an odd natural number then the dihedral group $D_{2n}$ is isomorphic to the group $D_n\times \mathbb{Z}_2$ where $D_m$ is the dihedral group of order $2m$, which has ...
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Finding $N(D_{4})/D_{4}$ for $D_{4}$ in $D_{16}$

I want to find $N(D_{4})/D_{4}$ where $N(D_{4})$ is the normalizer of $D_{4}$ in $D_{16}$. I'm not too clear on what the normalizer of $D_{4}$ in $D_{16}$ Is there a nice way to find $N(D_{4})/D_{4}$?...
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How to write dihedral group in cycle notation?

Since each symmetry can be thought of as a permutation of the vertices, the elements of $D_n$ can be thought of as elements of $S_n$. So I'm wondering if there's a systematic way that we can always ...
About proving that $\operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n)$ [closed]
How can I prove that $$\operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n),$$ where $\mathbb {D}_n$ is the dihedral group. Can someone help me please? ...