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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.

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What are dihedral groups?

I have trouble understanding what exactly a dihedral group is. I read about how they are rotations and reflections along the faces of a polygon. But then what does that mean? Whatever you do to a polygon you get the same polygon right? Could someone…
user2277550
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Isomorphism between dihedral group and a subgroup of $S_n$

I need to find an isomorphism between $D_n$ (all symmetries of an $n-gon$) and a subgroup of $S_n$. I know that Cayley's theorem gives a nice isomorphism that shows that $D_n$ is isomorphic to a subgroup of $S_{2n}$, but this isn't the case here,…
so.very.tired
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Expressing a product in a Dihedral group

Write the product $x^2yx^{-1}y^{-1}x^3y^3$ in the form $x^iy^j$ in the dihedral group $D_n$. I used the fact that the dihedral group is generated by two elements $x$ and $y$ such that: $y^n=1$, $x^2=1$ and $xy=y^{-1}x$ and I found that…
amir
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Understanding the role of vertices in $D_{2n}$ in the definition of rigid motion

I'm trying to understand a paragraph from Keith Conrad's expository paper on the dihedral group. The paragraph is: Pick two adjacent vertices of a regular $n$-gon, and call them $A$ and $B$ as in the figure below. An element $g$ of $D_n$ is a rigid…
Cardinality
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Why a dihedral group isn't a normal subgroup of a permutation group?

A claim says that two permutations are conjugate when they have the same type. I don't know how to show (using this claim) that a dihedral group $D_{2n}$ isn't a normal subgroup of a permutation group $S_n$. Thank you.
Cygne
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Group Theory - Dihedral Groups

Two questions related to Dihedral groups: What is the conventional notation for Dihedral groups? Is it Dn where n is the number of sides in a regular n-gon, or is it D2n where n is the number of symmetries in the n-gon? AQA haven't specified for…
user634745
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Show that the dihedral group $D_6$ of order $12$ has a nonidentity element $z$ such that $zg = gz$ for all $g ε D_6$.

From notes, I think all of the following are true: Every element of $D_6$ can be written as $s^ir^j$, where $i = 0,1$ and $0\le j\le 5$. $r^6 = e$, where $e$ is the identity. $s^2 = e$ $r^ks = sr^{-k}$ for any integer $k$. Do I actually have to…
Chris
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