# 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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### How to show that $|D_{2n}| = 2n$ via the presentation?

Consider the dihedral group $$D_{2n}= \langle a,b \mid a^n = 1 = b^2, b^{-1}ab = a^{-1}\rangle$$ How can I show that $|D_{2n}| = 2n$? I'm trying to show that we can write every element in the form ...
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### Extension of dihedral group to higher dimensions

The dihedral group $D_{2n} = \{x, y \mid x^2=y^n=yxyx=1\}$ is tied with the symmetries of the regular polygon on a plane. What is the natural extension to higher dimension? For instance, in $3$D, does ...
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### Symmetries of a regular tetrahedron

Let $G$ be the group of symmetries of a regular tetrahedron $T$, including orientation-reversing symmetries. (a) Decompose the set of faces of $T$ into orbits, and describe the stabiliser of a face. ...
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### Problems related to symmetry

I am currently studying the chapter entitled "Symmetry" from Michael Artin's book "Algebra" and am having some difficulties understanding the material. It is dealing with isometries, dihedral groups, ....
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### Is $G$ isomorphic to the dihedral group $D_{10}$?

Let $G\le S_6$ be the subgroup generated by the permutations $\sigma=(12356)$ and $\tau=(26)(35)$. I'm asked to determine: (a) the order of $G$ and the period of each element; (b) if $G$ is ...
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### How many distinct composition series does the group $D_{12}$ have?

How many distinct composition series does the group $D_{12}$ have? I know that $D_{12} \trianglerighteq \mathbb{Z}_6 \trianglerighteq \mathbb{Z}_3 \trianglerighteq \{e\}$ is a composition series ( ...
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### Dihedral group of order 10 in GAP

The dihedral group of order $10$ is given by $D_{10} = \langle a,b| a^5 = b^2 = 1, bab^{-1} = a^{-1}\rangle$. Now I need to find all the elements in GAP. But whenever I type ...
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### Dihedral group generated by $\langle r,s\rangle$ for all $n$

Under wikipedia for Dihedral groups it claims the following: The $2n$ elements in $D_n$ can be written as $\{e,r,r^2,r^3,\ldots,r^{n-1},s,rs,r^2s,\ldots,r^{n-1}s\}$. I know why this is true and it ...
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### Find $G/Z(G)$ given the following information about the group?

$G$ is a finite group generated by two elements $a$ and $b$, we are given the following data: Order of a= $2$ Order of $b=2$ Order of $ab=8$. If $Z(G)$ denotes the center then what is $G/Z(G)$ ...