# Tagged Questions

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

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### Normalizer of a Sylow 2-subgroups of dihedral groups

I can't solve the following exercise which is the last exercise in page 146 of Dummi & Foote's Abstract Algebra: Let $2n=2^ak$ where $k$ is odd. Prove that the number of Sylow 2-subgroups of ...
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### How can I show that $D_{2n}$ follows from these relations?

Suppose we have a group $A$ which is generated by generators $R$ and $F$, subject to the relation $$R^n=I, F^2=I,RF= FR^{-1}.$$ It should be just the dihedral group of order $2n$, the one generated ...
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### Why Composition and Dihedral Group have reverse order of operation?

NOTE - I didn't receive any answer in here and I think because my first post is not clear, so I entirely made another example: $K={\{id,r^2,r^4,s,r^2s,r^4s}\}$ is a proper subgroup of the dihedral ...
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### $x,y \in G$, $o(x) = 2, o(y) = 2, o(xy) = k, k\geq 3$. Show that $D_k \cong \langle x,y\rangle$

I know that $D_k = \{1,r, \dots, r^{k-1}, s, sr, \dots, sr^{k-1} \}$ and that I can use $\phi: s^i r^j \mapsto x^i(xy)^j$. I don't know what can be found using this.
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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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### Dihedral subgroups of $S_4$

Prove that in $S_4$ there are $3$ groups that are isomorphic to $D_4$. I know that the $2$-sylows of $S_4$ should be subgroups of order $8$, but to prove it is a bit tricky for me Any help ...
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### Dihederal Group $D_{2n}$ Where $n$ is even/odd

I know that the group presentation of $D_{2n}$ is the following $$D_{2n} = \big<a,b: a^n=b^2=1,b^{-1}ab =a^{-1} \big>$$ Now if we consider the case where $n$ is even and we write $n =2m$ for ...
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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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### 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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### In how many different ways can one color the vertices of a regular pentagon into four colors?

I am trying to find the number of ways to color a pentagon with 4 colors up to symmetries. I know that I should be using Burnside's Theorem, and so far I know that the group $D_5$ should act on the ...
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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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### Is every symmetric group generated by some cycles?

Actually, I have two questions here: How to show that $D_3$, the dihedral group, is isomorphic to $S_3$ in details? Is every symmetric group generated by some cycles? For question (1), I know ...
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### The Dihedral group $D_1$ is non-abelian?

Same as above. I'm trying to show that for any n being odd, $D_n$ has exactly n elements of order 2 where $D_n$ is non-abelian. I know that for $n\ge3$ this is true, but what about for $n=1$.
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### Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
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### Proof that $S_3$ isomorphic to $D_3$

So I'm asked to prove that $$S_{3}\cong D_{3}$$ where $D_3$ is the dihedral group of order $6$. I know how to exhibit the isomorphism and verify every one of the $6^{2}$ pairs, but that seems so long ...
### 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 ...
### Why does $\rho_{\mathbf{Z}^t\otimes P}=\rho_R$ imply the isomorphsim of $\mathbf{Z}^t\otimes P\cong R$?
so I have happened upon a thesis regarding the calculations of various $\mathbf{Z}D_6$ modules and their isomorphisms and came across a technique which is bothering me. Let me give an example. Let ...