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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

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Show that there exist an injective homomorphism of dihedral group $D_n$ into $G$.

Let: $$n\gt2 \; \text{and the group} \; (G,⋅)$$ Consider that there existe: $a,b∈G$ such that $a^n=b^2=1_G$ and $b⋅a=a^{−1}⋅b$ and $n$ is the smallest $n≥1$, such that $a^n=1_G$. ...
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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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Are all abelian subgroups of a dihedral group cyclic?

Are all abelian subgroups of a dihedral group cyclic? Attempt: I have counter-examples for n=1,2 so I know that it isn't true for n<3. Is it true for n≥3? How do you know this?
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$D_3\oplus D_4$ not isomorphic to $D_{24}$

We need to prove that $D_{3} \oplus D_{4}$ is not isomorphic to $D_{24}$ . The way in which I approach such type of questions is to count the number of elements of order $x$ in one group and then in ...
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Group Theory- symmetry group

The Symmetric group of the set ${1,2...,n}$ is $S_n$. It is the set of permuations of the set ${1,2...,n}$. But what is the symmetry group of a polygon? or a $3D$ shape? For example I saw: "A ...
907 views

A group is generated by two elements of order $2$ is infinite and non-abelian

My question is as follows: Let $G = \langle a,b \mid a^2=b^2=1 \rangle$ be a group generated elements $a, b$ and the equation $a^2=b^2=1$. Prove that $G$ is infinite and non-abelian. I got the ...
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How shall we show that the only possible orders of any element in dihedral group $D_n$ will be either a divisor of 2 or $n$?

I am studying the dihedral group $D_n:=\{r_n, f_n: r_n^n=f_n^2=(r_nf_n)^2=e_n\}$. I am willing to show that the possible orders of any element in it will be either a divisor of 2 or $n$. But don't ...
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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 ...
76 views

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 would be ...
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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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Why are automorphisms of $D_{2n}, n \geq 5$ odd, not always inner?

The dihedral group of order $2n$ is often presented as $$D_{2n}= \langle r,s: r^n=s^2=1, rs= sr^{-1} \rangle \text{,}$$ where $r$ denote rotations and $s$ denote reflections of a regular $n$-gon. ...
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Dihedral group of order 2n

I would appreciate if someone could prove this for me: Let G be a dihedral group of order 2n and suppose H is a cyclic quotient group of G. Show that |H|is less than or equal 2.
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Find a composition series of $D_8$

Answer: Let $D_8=\langle a,b\rangle=\{e,a,a^2,a^3,b,ba,ba^2,ba^3\}$, where $a^4 = b^2 = e$ and $ab = ba^{-1}$ as usual. Then $$\{e\}\lhd\langle a^2\rangle\lhd\langle a\rangle\lhd D_8$$ or \{e\}\...
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Find all $x$ in $D_4$ such that $ax(a^{-1}) = b$.

Consider the dihedral group $D_4$. Consider also the elements $a= r_1$ and $b= S_1$ of $D_4$. Find all $x$ in $D_4$ such that $ax(a^{-1}) = b$. Do both $a$ and ($a^{-1}$) cancel each other out? If not,...
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Find all the $p$-Sylow subgroups of $D_6$.

$|D_6|=12=2^23.$ I started with $3$. I know that the number of $3$-Sylow subgroups, denoted $n_3$, is: $1,4,7...$ and I also know that $n_3|2^2$. e.g, $n_3=1, 4$. How can I show that it can't be $4$? ...
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Commutator of $D_{2n}$

I am trying to calculate the commutator of the Dihedral group. If $n=1,2$ then $[D_n,D_n]=1$. Now I consider the case $n\geq 3$. I thought of using the property $[G,G] \subset H$ iff $H \lhd G, G/H$ ...
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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 ...
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 ...
|${G_x}$| 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 $D_{10}$ ...