# 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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### 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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### A group is generated by two elements of order $2$ is infinite and non-abelian

My question is as follows: Let $G = \langle a,b \mid a^2=b^2=1 \rangle$ be a group generated elements $a, b$ and the equation $a^2=b^2=1$. Prove that $G$ is infinite and non-abelian. I got the ...
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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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### Coloring sides of truncated triangular dihedral(bipiramid) into 3 colours

I need to find out the amount of ways to colour truncated triangular dihedron into 3 colours. So, the task will be easier if I had simple triangular dihedron. First of all, do I understand right ...
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### Dihedral group is supersolvable

I need to show that Dihedral group $D_n$ is supersolvable. My Approach : I think the existence of a normal chain $\{e\} = G_0 \leqslant G_1 \leqslant ... \leqslant G_n = G$ satisfying following ...
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### Irreducible representation of $C^*(D_\infty)$

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$ Ultimately, I'm interested in finding a ...
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### |${G_x}$| of ${D_{10}}$

I am looking for the order of the stabilizer group of $D_{10}$. I know that ${G_x} = \{g \in G : gx = x\}$. I am curious what to use for $x$ though? Should I just cycle through elements of $D_{10}$ ...
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### Automorphisms in the Dihedral groups

Let g be a group and $a \in G$. Define $\phi_a:G\rightarrow G$ by $\phi_a(g)=aga^{-1}.$ Now Let $G=D_4$ and $a=r$, where $r$is the rotation. We must show that $\phi_r: D_4\rightarrow D_4$. So show ...
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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?
### How to geometrically show that there are $3$ $D_4$ subgroups in $S_4$?
As shown in this note, the symmetry group $S_4$ for a cube has $3$ subgroups that are isomorphic to $D_4$, the dihedral group of order $2 \times 4 = 8$. How to geometrically illustrate this fact? ...
What notation is most common for the dihedral group of order $2n$? I'm talking about the group of symmetries of a regular $n$-gon. I know that some books call this group $D_n$, and some books call it \$...