# 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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### Can we build a D4 matrix representation with ${\mathbb Z_2}^{4\times 4}$ matrices?

If we look at the Dihedral 4 group. There exists a trivial matrix representation that also makes it very easy to define group action on vectors: simply use 2x2 rotation matrices for the cyclic part ...
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### How to prove isomorphism with the Dihedral group

I have a group that I'm trying to prove is isomorphic to the Dihedral group. I know that it is finite, that it is generated by two elements $\alpha$ and $\beta$ such that: $\alpha^2=\beta^n=1$ and ...
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### Images of the generators of $D_{10}$ under its automorphisms.

I have constructed the dihedral group generated by $a$ and $b$ of order $10$ in GAP by the following way: ...
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### understanding the commutator of dihedral group [duplicate]

Let $G=D_{2n}=⟨x,y|x^2=y^n=e,$ $yx=xy^{n-1}⟩$ I need to find $G'$ [ the commutator of G] now I understand that $G'$ is the subgroup generated from $U=xyx^{-1}y^{-1} ,$ $\ \forall x,y \in G$ So,...
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### Two questions about the dihedral group

First question: 1) Is the sum of subgroup indices of dihedral group with $2n$ elements equal to $\sigma_2(n)+2\cdot \sigma(n)$? Second question: 2) Is $\sigma_2(n)+2\cdot \sigma(n) \le L(H(D_n))$? ...
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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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### classification of representations of $D_{1009}$

A follow-up of this question To fix ideas, take $n=1009$. $D_n$ has $2$ irreducible representations of degree $1$ and $504$ representations of degree $2$. Are the degree 1 representations all ...
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### Orders of the Elements of $D_6/Z(D_6)$

I have been trying to calculate the orders of the elements of $D_6/Z(D_6)$. For example, using $R_{60}$ to represent rotation by 60 degrees and $R_0$ to represent rotation by 0 degrees (the identity ...
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### Question about conjugacy classes in dihedral groups [duplicate]

I'm trying to find the conjugacy class of a rotation $r^{k}$. Is it unitary? How about a symmetry $s$? Any ideas?
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### Why is $\langle r\rangle$ characteristic in $D_n$?

I need to determinate if $\langle r\rangle$ is characteristic in $D_n = \langle r \rangle_n \rtimes \langle s \rangle_2$. This is trivial if I use the result that every cyclic group is ...
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### Understand the free group universal property applied to $D_n$

For $n ≥ 3$ and $D_n$ the dihedral group of order $2n$ with présentation $\langle r, s : r^n = s^2 = srsr = 1\rangle$ prove that for all $(a, b) \in (\Bbb Z/n\Bbb Z)^2$, there exists a morphism $f$ ...
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### How to find number of abelian subgroups of diheral group? [closed]

How to find number of abelian subgroups of diheral group $D_n$? Attempt: I have counter-examples for $n=1,2$ so I know that it isn't true for $n<3$. Is it true for $n\ge 3$? How do you know this?...
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