Questions tagged [dihedral-groups]

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

385 questions
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Show every irreducible representation of $D_{n}$ must have dimension less than or equal to 2

This question was homework once upon a time. I have long since handed it in. "Let $D_{n}$ be the dihedral group with $2n$ elements. Show that every irreducible representation of $D_{n}$ must have ...
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Subgroup of Dn isomorphic to Q8

Is there an n such that $D_n$ contains a subgroup isomorphic to $Q_8$? My immediate thought is no, but I'm not sure how to prove it. I know that there are only 2 non-Abelian groups of order 8 (up to ...
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Show $T(x)=x+1$ and $I(x)=-x$ produce a dihedral group$G := D_n$

Lets look at a musical note system with $n$ notes. We see two notes as the same when they differ one octave. We write the collection of notes as $X= \Bbb Z_n$ $T: X \rightarrow X$, $T(x)=x+1$ ...
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Defining dihedral groups $\{\sigma \in S_n:$ something $\}$

I am trying to understand hos one can define the dihedral groups $D_n$. I have seen the "definition" that just says this is the group of symmetries of an $n$-polygon. So you have rotations and ...
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Tricky question on Dihedral groups [duplicate]

If $$D_{2n} = \{r^i s^j, \quad 0 ≤ i ≤ n − 1, \quad 0 ≤ j ≤ 1 \}$$ is the dihedral group of order $2n$, show that the center of $D_{2n}$, $n > 2$ is the trivial subgroup if $n$ is odd, and is the ...
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What does “A function on the $n-$gon” mean?

I am reading Dummit and Foote. We have: 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 $n-$gon which ...
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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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Expressing dihedral group as internal direct product of normal subgroups

Can dihedral groups be expressed as internal direct product of two normal subgroups? I think no, since an element of a normal subgroup must commute with every other element of other normal subgroup, ...
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Why are Transpositions and rotations in dihedral groups the same operation?

A dihedral group seems to have 2 operations, rotation, and transposition. These transformations seem so distinct, no composition of rotations may ever lead to a transposition and vice versa (right?). ...
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Help with understanding proof: every group of order 6 is cclic or dihedral.

I am considering the proof that every group of order 6 is cyclic or dihedral (following some lecture notes). The initial part of the proof, in broad brushstrokes, considers that the elements in the ...
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I want to make sure I am correct about something I read in Moore and Pollatsek's "Difference Sets": Suppose we have an abelian group $H$. If $\exists g\notin H$ with $g^2=1$ and $ghg^{-1}=h^{-1}\;... 2answers 78 views Dihedral groups and reflections Let$\rho\in D_{10} $be a rotational symmetry in$\frac{2\pi}{5}$radians non-clockwise,and let$\epsilon\in D_{10}$be a reflection symmetry (related to the X-axis).Prove that every$\epsilon\rho^i\...
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 ...
It's well-known that, for any $n$, we can consider $D_n = \langle r, f | r^n = e, f^2 = e, r^k \not = e (0 < k < n), fr= r^{-1} f\rangle$; However, not all two element generating sets for \$...