# Questions tagged [dihedral-groups]

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

55 questions
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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 ...
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### Subgroups of $D_4$

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..
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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, ...
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### Some Subgroup of Dihedral Group is Normal

I ran into this question when I was studying for my abstract algebra midterm. Show that the subgroup $H$ of rotations is normal in the dihedral group $D_n$. Find the quotient group $D_n/H$. I'm ...
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### Center of $D_6$ is $\mathbb{Z}_2$

The center of $D_6$ is isomorphic to $\mathbb{Z}_2$. I have that $$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$ $$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$ My ...
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### $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$

Problem statement: I need to find out if $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$ and explain my answer. I already know that it is isomorphic, so now all I need to do is to prove it. I assume that ...
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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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### Dihedral Groups; what exactly are the elements of the set?

I am reading Dummit and Foote. We have: [Definition 1:]For each $n \in \mathbb{Z}^+, n \ge 3$ let $D_{2n}$ be the set of symmetries of a regular $n-$gon, where a symmetry is any rigid motion of the ...
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### How to show $D_3\oplus D_4$ is not isomorphic to $D_{24}$? [duplicate]

How to show $D_3\oplus D_4$ is not isomorphic to $D_{24}$? Here $D_n$ is the dihedral group of order $2n$. I am not sure how to prove this. I am not very good with the dihedreal groups.
This question is supplementary to another question. From that question, we know that the automorphism group of the $N$ disjoint cycle graphs of same length $n$ is $S_N \wr D_n$. My question: What is ...
### Is $D_{2n}$ isomorphic to $D_n \times \Bbb{Z}_2$ for all $n$? For all odd $n$? [duplicate]
Is $D_{2n}$ isomorphic to $D_n \times \Bbb{Z}_2$ for all $n$? For all odd $n$? I just want to see if my thinking is sound here. My thought process is this. $\mathbb{Z}_2 \cong \{e,j\} \subset D_n$ ...