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Questions tagged [dihedral-groups]

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

In mathematics, a dihedral group, is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. It is well-known and quite trivial to prove that a group generated by two involutions is a dihedral group.

754 questions
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Proof that $S_3$ isomorphic to $D_3$

So I'm asked to prove that $$S_{3}\cong D_{3}$$ where $D_3$ is the dihedral group of order $6$. I know how to exhibit the isomorphism and verify every one of the $6^{2}$ pairs, but that seems so long and tedious, I'm not sure my fingers can…
Addem
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I need to determine the subgroups of the dihedral group of order 8, $D_4$.

I need to determine the subgroups of the dihedral group of order 4, $D_4$. I know that the elements of $D_4$ are $\{1,r,r^2,r^3, s,rs,r^2s,r^3s\}$ But I don't understand how to get the subgroups..
amir
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Group cohomology of dihedral groups

If $m$ is odd, the group cohomology of the dihedral group $D_m$ of order $2m$ is given by $$H^n(D_m;\mathbb{Z}) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/(2m) & n \equiv 0 \bmod 4, ~ n > 0 \\ \mathbb{Z}/2 & n \equiv 2 \bmod 4 \\ 0 & n \text{…
Martin Brandenburg
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The smallest symmetric group $S_m$ into which a given dihedral group $D_{2n}$ embeds

Several questions, both here and on MathOverflow, address the issue of determining for a given group $G$ the smallest integer $\mu(G)$ for which there is an embedding (injective homomorphism) $G \hookrightarrow S_{\mu(G)}$. In general this is a…
Travis Willse
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Is the dihedral group $D_n$ nilpotent? solvable?

Is the dihedral group $D_n$ nilpotent? solvable? I'm trying to solve this problem but I've been trying to apply a couple of theorems but have been unsuccessful so far. Can anyone help me?
nomadicmathematician
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Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$.

Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$. I define a map $$\phi:D_{2n} \rightarrow \mathbb{Z}_2 \times D_n$$ by $R \mapsto (0,r^{\frac{n+1}{2}})$ and $M \mapsto(1,m)$. Then I am stuck at showing the map is…
Idonknow
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Looking for a simple proof that groups of order $2p$ are up to isomorphism $\Bbb{Z}_{2p}$ and $D_p$ for prime $p>2$.

I'm looking for a simple proof that up to isomorphism every group of order $2p$ ($p$ prime greater than two) is either $\mathbb{Z}_{2p}$ or $D_{p}$ (the Dihedral group of order $2p$). I should note that by simple I mean short and elegant and not…
Serpahimz
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Does there exist a group $G$ such that $\operatorname{Aut}(G)\cong D_5$, where $D_5$ denotes the dihedral group of order 10?

I came across this problem stated in the title having no clue what to do, and got stuck even in the finite case. Here's my attempt: I first proved that the group $G$, if it exists, cannot be finite abelian. If it were, we could factor $G$ into…
Cyankite
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Field extension with dihedral Galois group

In an old exam of my Galois Theory class there is the following question which troubles me: Let $p \neq 2$ be a prime number and $k \geq 1$ an integer. Give an example of a galois extension $L/K$ such that $Gal(L/K) = D_{2p^k}$ and…
Zorba le Grec
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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 (1918), $n$ can be partitioned in…
Omar Shehab
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Finding Sylow 2-subgroups of the dihedral group $D_n$

I am trying to describe the Sylow $2$-subgroups of an arbitrary dihedral group $D_n$ of order $2n$. In the case that $n$ is odd, $2$ is the highest power dividing $2n$, so that all Sylow $2$-subgroups have order $2$, and it is fairly easy to…
KyleP
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Prove that the dihedral group $D_4$ can not be written as a direct product of two groups

I like to know why the dihedral group $D_4$ can't be written as a direct product of two groups. It is a school assignment that I've been trying to solve all day and now I'm more confused then ever, even thinking that the teacher might have missed…
harmajn
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Center of dihedral group

I am trying to solve the following exercise about the dihedral group and its center: If $g\in Z(D_{2n})\Leftrightarrow ga=ag, bg=gb$, where $a,b$ are generators of $D_{2n}$. We have defined the dihedral group of order $2n$ as $D_{2n}=\left \{…
Lullaby
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Subgroup structure of dihedral groups

I am currently looking into structure of dihedral groups; I am interested in their subgroup structure. Dihedral group have two kinds of elements; I will use their geometric meaning and call them rotation and reflection: $r$ and $s$. All dihedral…
Dark Archon
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Confusion about centraliser of $D_5$ in $GL_4(\mathbb{Z})$

I am trying to follow a derivation on a very old paper. My knowledge of group theory is limited, I have the basis but not much experience with advanced concepts. We are working in 4 dimensions, so the paper quotes the 4D representations of the…
SuperCiocia
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