# 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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### Proper condition on the dihedral group

Is there a theream which is a condition on $n\in\mathbb N$ that says when the dihedral group, $D_{n}$, has non-cyclic subgroups? After spending some time figuring a condition I tried to find some ...
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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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### Alternative way of proving the subgroup of rotations is normal in $\mathbb D_4$

I've just solved a basic group theory exercise which is: decide if $\{1,r,r^2,r^3\}$ is a normal subgroup of $\mathbb D_4$ (I mean the dihedral group of $8$ elements, not the one of $4$). I've used ...
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### Orders of the Elements of $D_6/Z(D_6)$

I have been trying to calculate the orders of the elements of $D_6/Z(D_6)$. For example, using $R_{60}$ to represent rotation by 60 degrees and $R_0$ to represent rotation by 0 degrees (the identity ...
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### Element not in the subgroup $<x>$ of the dihedral group is a reflection

The dihedral group $D_{2n}$ is generated by $x$ and $y$ such that $x^n = y^2 = xyxy = e$. Show (algebraically) that elements not in the subgroup $<x>$ is a reflection and find the line (...
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### Dihedral Group question involving $D_{4n}$

Let $D_{4n}$ be the dihedral group of the order $4n$. Prove that $D_{4n}/\langle T^n \rangle$ is isomorphic to $D_{2n}$. We tried to configure an action on the diagonals of the $n$-gon and prove ...
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### How many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads?

I am trying to figure out how many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads. I understand how to do the problem with two colors, but I am struggling to ...
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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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### Given a Cayley table, is there an algorithm to determine if it is a dihedral group?

Showing that it is a group is simple enough, but is it possible to determine if it is a dihedral group or not just by looking at the Cayley table?
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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 ...
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### Dihedral group as a direct product

In a problem from a past exam I am asked "When can $D_n = \langle r,s\mid r^n = s^2 = (rs)^2 = 1\rangle$, the dihedral group of order $2n$, be expressed as a direct product $G\times H$ of two ...
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### Find the inner and outermorphisms of a particular dihedral group

Given that |Inn($D_8$)| = 8 and |Out($D_8$)| = 2 where Out($D_8$) = Aut($D_8$)/Inn($D_8$) and $D_8$ = {e,r,$r^2$,..,$r^7$,s,sr,...,$sr^7$} we want to find Inn($D_8$) and Out($D_8$). We know that Out(...
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### How many homomorphisms are there from $D_5$ to $V_4$?

Question: How many homomorphisms are there from $D_5$ to $V_4$, where $D_5$ is the dihedral group of order $10$ and $V_4$ the Klein four-group? I've used the fact that since $V_4$ is abelian, the ...
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### In dihedral group $FR=R^{-1}F$

Let F be any reflection (flip ) about axis of symmetry . And R be rotation by $\frac{2\pi}{n}$ radian counterclockwise .(n is the number of vertex). Then $FR=R^{-1}F$ I looked at some example and ...
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### Determining order of elements and number of automorphisms in dihedral groups

Reviewing some stuff and found myself confused at a few things involving dihedral groups and automorphisms, would very much appreciate some assistance in understanding. Namely beginning with this, I ...
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### Orbits of rotations under action by D4.

If D4 is acting on the subgroup of its rotations, C4, by conjugation, what are the orbits? I believe that the orbit of each rotation is itself and its own inverse rotation and nothing else. For the ...
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### Show that the additive group $\mathbb R$ acts on the $x, y$ plane $\mathbb R \times \mathbb R$ by $r\cdot (x, y)=(x+ry,y)$.

Show that the additive group $\mathbb R$ acts on the $x, y$ plane $\mathbb R \times \mathbb R$ by $r\cdot (x, y)=(x+ry,y)$. I am completely lost with this one partly because I do not understand group ...
### 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? ...
$D_4$ is generated by a rotation $\alpha$ of order 4 and a reflection $\beta$. Its elements $e$, $\alpha$, $\alpha^2$, $\alpha^3$, $\beta$, $\alpha\beta$, $\alpha^2\beta$, $\alpha^3\beta$ give an ...