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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How to formally derive semidirect product for D4 group?

I am fairly new to Group Theory. I know that $D_4$ is a combination of $C_4$ and $C_2$. Now, how to derive semidirect product for D4 group? For instance, in $P_4$ group, if $t, p$ are translations, ...
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What is actually the center of semi dihedral group $SD_{24}$?

I got confused whether the center of semi dihedral group $SD_{24}$ is just ${e,a^6}$ or ${e,a^3,a^6,a^9}$ since all of elements are commute and it also equals to its inverse. My definition for semi ...
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Understanding von Dyck's theorem

I'm trying to understand how to use Von Dyck's theorem to prove that $S_3 \cong D_3$. I believe I have a correct sketch, but I'm very fuzzy on the details, mainly because I haven't seen free groups ...
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Proof idea: $S_3 \cong D_3$ [duplicate]

I am trying to show that $S_3$ is isomorphic to $D_3$ as groups. The definitions I'm working with are $$ S_3 = \left \langle (12), (123) \right \rangle, \; D_3 = \left \langle r, s \mid r^3 = s^2 = 1, ...
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Any finite subgroup of $I(\mathbb{R}^2)$ (isometry group of $\mathbb{R}^2$) is either cyclic or dihedral.

I'd like to show it by counting the number of all fixed points. To be specific, i) If a finite subgroup $G$ only fixes one point in $\mathbb{R}^2$, ii) If there are at least two fixed points. We ...
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Formula for the number of ways to choose $n$ digits from $k$ possible digits such that the selection contains no repeats under permutation

An example to illustrate: Say I have 3 digits, to be taken from the set $\{1,2,3\}$. I want to choose from this set of digits, with repetition allowed, to make triplets. However, no two triplets ...
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Let $n$ be an even number. Prove that $D_n/Z(D_n)$ is isomorphic to $D_{n/2}$.

Let $n$ be an even number. Prove that $D_n/Z(D_n)$ is isomorphic to $D_{n/2}$. To avoid confusion say: $$D_n=\{a^ib^j:i\in\{0,1\}, j\in\{0,\ldots,n-1\}, ba=ab^{-1}\}.$$ Through some calculation we ...
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How would you describe the symmetries/group actions on the $\ell_p$ circle?

We define the unit circle as the collection of all vectors with length 1 centered at some point. (The one below specifically defines the unit circle centered at the origin) $$\mathscr{C} = \{ x\in\...
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Dummit & Foote Chapter 1.2 Exercise 17 (Group Presentations)

The exercise is to show that if $n = 3k$, then the group presentation $\langle x, y \mid x^n = y^2 = 1,\ xy = yx^2 \rangle$ generates $D_6$. The group presentation for $D_6$ given in the book is $D_6 =...
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Question about center of dihedral group [duplicate]

I need suggestions for this exercise. Let $D_{2n}$ stand for the dihedral group of $2n$ elements. Show that if $n$ is odd and $\forall b \in D_{2n} (ab = ba)$, then $a = e$. Show that if $n$ is even, ...
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Check if kernel and image for dihedral group defined correctly

Suppose that $n = dm$ where $d$ and $m$ are positive integers with $m\ge 3$. Consider the dihedral group $D_n = \langle \{\mu, \rho\}\rangle,$ where $|\mu| = 2$, $|\rho| = n$ and $\rho\mu = \mu\rho^{−...
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Show the function between the dihedral groups is well defined

Suppose that $n = dm$ where $d$ and $m$ are positive integers with $m\ge 3$. Consider the dihedral group $D_n = \langle \{\mu, \rho\}\rangle,$ where $|\mu| = 2$, $|\rho| = n$ and $\rho\mu = \mu\rho^{−...
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Question on the decomposition of a representation of $D_n$ into irreducible representations.

For simplicity I‘ll just consider the case in which n is even. $D_n$ is given by $\langle r,t : r^n=s^2=1, (rt)^2=1\rangle $ we can construct a representation on $\mathbb{C}^n$ $\rho: D_n \...
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When is $D_n\approx\operatorname{Aut}(D_n)$? [duplicate]

We define the group $D_n$ to be the dihedral group of order $2n$ (equivalent to the group of rotations and reflections on a regular $n$-gon) and $\operatorname{Aut}(G)$ to be the group of ...
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Proof verification of Dummit and Foote exercise 1.6.24

Problem 1.6.24 in D&F: Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $n = |xy|$. My proof: If $t = ...
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Why must a group with presentation $\langle r,s \mid r^n=s^2=1,rs=sr^{-1}\rangle$ have exactly $2n$ elements?

In Dummit and Foote's Abstract Algebra, they state on page 27 of chapter 1 that any group with the presentation $\langle r,s \mid r^n=s^2=1,rs=sr^{-1}\rangle$ must have exactly $2n$ elements. Why is ...
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Let $G=D_8\times\mathbb{Z_6}$ and $N=D_8\times\{0\}$. Prove that $G/N$ is isomorphic to $\mathbb{Z}_6$.

Let $G=D_8\times\mathbb{Z_6}$ and $N=D_8\times\{0\}$. Prove that $G/N$ is isomorphic to $\mathbb{Z}_6$. My attempt: I proved that $N$ is a normal subgroup of $G$. Then we can define $$G/N = \{gN \mid ...
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Find the number of conjugacy classes of elements of order $2$ in $D_{2p}$ where $p$ is odd

Find the number of conjugacy classes of elements of order $2$ in $D_{2p}$ where $p$ is odd So I am trying to solve this problem, and I know that if $p$ is odd and $$D_{2p}=\langle r, s \mid s^2=1, r^...
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Why not $D_8/\langle r^2 \rangle = M?$

Source: Dummit and Foote book. Page number: $85$. It is written that $D_8/\langle r^2 \rangle = \{ x \{ 1, r^2\} : x \in D_8 \}=\{ \{1,r^2\} , \{ r,r^3\} , \{s, sr^2\} , \{sr, sr^3\}\} $ My ...
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Prove that the centralizer $C_{D_8}(A)=A$.

Let $D_8$ be the dihedral group of order $8$. Using the generators and relations, we have $D_8=⟨r,s∣r^4=s^2=1,sr=r^{−1}s⟩$. (a) Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{ 1,r,r^...
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How can I enumerate D6-symmetric cellular automaton rules?

I'm experimenting with rules for a cellular automaton in a hexagonal grid. I am wondering how to enforce symmetry. For example a cell can be either "alive" or "dead". Let's say a ...
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Is the infinite dihedral group isomorphic to $\mathbb{Z}*\mathbb{Z}$?

Let $a,b$ denote the generators of the copies of $\mathbb{Z}_2$ in the free product $\mathbb{Z}_2*\mathbb{Z}_2$. The infinite dihedral group is described by: $$D_{\infty} = \; \left<r,s \;|\; srs=r^...
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Prove that the groups $D_6$ and $\mathbb{Z}_6$ are isomorphic

I'm stuck trying to prove that: Prove that the groups $D_6$ and $\mathbb{Z}_6$ are isomorphic. My attempt: Let $D_6=\{e,a,a^2,b,ab,a^2b\}$ where $a^3=b^2=e$ and $ba=a^2b$ and $\mathbb{Z}_6=\{[0],[1],[...
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Each extra-special group of order $2^{2n+1}$ is a central product of $D_8$s or of $D_8$s and a single $Q_8$.

This is Exercise 5.3.7(i) of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to this search, it is new to MSE. This is a classification problem. This Wikipedia ...
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Show that $D_4$ is isomorphic to the direct product of two cyclic groups of order 2.

What I have tried is the following: Since $D_4$ is a group of order four, then it is abelian, so we use the fundamental theorem of abelian groups which tells us that every finite abelian group is ...
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2 answers
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Showing ${\rm Aut}(D_{2^n})\cong{\rm Aut}(Q_{2^n})$ for $n\ge 4$.

In this comment from back in 2013, it is claimed that $${\rm Aut}(D_{2^n})\cong{\rm Aut}(Q_{2^n})$$ for $n\ge 4$, where $$D_{2^n}\cong \langle r,s\mid r^{2^{n-1}}, s^2, srs=r^{-1}\rangle$$ is the ...
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Find the number of $2$-Sylow of $D_{10}$

Find the number of $2$-Sylow of $D_{10}$ I got this problem of group theory. I tried to solve it this way: Using third Sylow theorem, we know that the number of $5$-Sylow is $1$, so we have only one ...
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Image and kernel of the action of the group $D_{12}$ on the $D_{12}/H$

Let $D_{12}=\langle(1,2,3,4,5,6),(2,6)(3,5)\rangle$ and $H=\langle(1,4)(2,5)(3,6)\rangle$. Then $|H|=2$. I want to determine kernel and image of the action of $D_{12}$ on $D_{12}/H$ (set of right ...
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3 votes
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Finding example of a particular group

Give an example of a group $G$ of order $p^2q$ where $p$ and $q$ are distinct prime numbers satisfying $q\not\equiv 1 \pmod{p}$ which does not have a unique $q$-Sylow subgroup. My attempt: Consider ...
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Explicit examples of oracle in Dihedral Hidden Subgroup Problem

In the general hidden subgroup problem we are given a generating set of a group $G$ (not necessarily abelian), we are given access to a function $f:G\to\mathbb{C}$, such that there is an unknown ...
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The group ring of the dihedral group $D_6$

I'll use the $D_6=\langle f,r\mid f^2,r^3,frfr\rangle$ definition for $D_6$. I'm trying to study the structure of $\mathbb Z[D_6]$. Units One thing I've noticed $(1+fr)^2=1+2fr+(fr)^2=2(1+fr)$, ...
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Working out cyclic subgroups of $D_{8}$ [duplicate]

I've been trying to work out the cyclic subgroups of $D_{8}$ out by hand and thought I could do it but I've run into something I don't understand. $D_{8}$ = $\{1,r,r^{2},r^{3},s,sr,sr^{2},sr^{3}\}$ I ...
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Two dimensional representation of $D_5$

I am trying to figure out the $2$-dim representation of $D_5$. Consider the action of $D_5$ on $\mathbb{R}^2$ by rotation and reflection. Then we can define a $2$-dim representation $\rho:D_5\to GL_2(\...
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Non-trivial blocks of order $16$ of $D_{16}$ acting on $8$-gon vertices

I would like to know exactly how many non-trivial blocks has $D_{16}$ of order $16$ acting on $8$ vertices of $8$-gon. I guess I should do it taking blocks as unions of the orbit of stabilisers but I ...
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Show that there exists a monomorphism from $D_n$ to $S_n$.

Problem: Show that there exists a monomorphism from $D_n$ to $S_n$, $n\geq 3$. Write $D_n=\langle x,y\mid x^n=1, y^2=1, yx=x^{-1}y\rangle$. Define a map $\phi:D_n\to S_n$ such that $ \phi(x)=\begin{...
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Show that $D_n$ is a group with composition.

Problem: Show that $D_n$ is a group with composition, where order of $D_n$ is $2n$ and $$D_n=\{e,a,...,a^{n-1},b,ab,...,a^{n-1}b\}$$with the following relations: $a^n=e$, $b^2=e$, $ba=a^{n-1}b$. ...
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Providing an example of a group $G_1$ and a group $G_2$

Can someone please help me with the last part of this problem I am working on? All my work is below. Thank you for your time and help. Let $N = 10m,$ where $m> 1$ and $5\nmid m.$ i. Let $G_1$ be a ...
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Subgroups of $D_4$

I'm trying to find the subgroups of $D_4$, the group of symmetries of the square. One way this could be done is to go through each element and every possible subset and check the axioms: the identity ...
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Accessing Elements of Group in GAP [closed]

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Show that there are exactly 4 homomorphisms from $ D_4 $ to $ Q_8$

I have a task where I need to show that there are exactly 4 homomorphisms from $D_4 $ to $ Q_8 $. I've been trying to use the fact that if $ \varphi: G \rightarrow H $ is a homomorphism, then $ |\...
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2 votes
1 answer
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How to visualise $1$-gon and $2$-gon for the dihedral groups $D_1$ and $D_2$?

For $n≥3$, the Dihedral groups may be defined as a collection of rotational isometries $r, r^2, \ldots, r^n=e$ and reflection isometries $s, sr, sr^2, \ldots, sr^{n-1}$ satisfying $(sr)^2 =e$ of $n-$...
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How are dihedral groups $D_1$ and $D_2$ defined.

I'm a beginner here so I'd really appreciate an answer in simple English. Wikipedia defines Dihedral groups as a collection of rotational isometries $r, r^2, \ldots, r^n=e$ and reflection isometries $...
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Defining Dihedral groups using reflections.

I'm trying to understand Dihedral groups $D_n$ of order $2n$. I'm a beginner at Group theory and my textbook uses Dihedral groups to motivate Group Theory so I haven't really began studyng group ...
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If two lattices have the same kissing number and center density, then are they similar?

Two lattices $\Lambda$ and $\Omega$ (the $\mathbb{Z}$-span of a linearly independent set $B\subset \mathbb{R}^n$) are said to be similar if there exist a real orthogonal $n \times n$ matrix $A$ and a ...
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1 vote
3 answers
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Show that dihedral group of twice odd order doesn't have a normal subgroup of order $2m$, where $m$ divides $n$.

Let $n \ge 3$. Let $D_n = \langle r,s \rangle$, for $r^n=s^2=1$ and $rs=sr^{-1}$. Then $D_n$ (not $D_{2n}$) is the dihedral group of order $2n$. Show that if $n$ is odd, then $D_n$ doesn't have a (...
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2 answers
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Prove dihedral group has subgroup of order $m$ and then of order $2m$, where $m$ divides $n$.

Let $n,m$ be integers with $n \ge 3$ and $m \ge 1$ and where $m$ divides $n$. Prove that the dihedral group of order $2n$, denoted $D_n$ (instead of $D_{2n}$), has subgroup of order $m$ and then of ...
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Is order of dihedral group part of the definition? Or something to be proven?

Re the close vote: There are only 2 questions here. 1 main question. 1 side question. I'm trying to understand what's assumed vs proven in dihedral group. In particular I'm trying to understand ...
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1 vote
1 answer
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Determine whether $D_4/Z(D_4)\otimes Z(D_4)$ is isomorphic to $D_4$

Determine whether $D_4/Z(D_4)\otimes Z(D_4)$ is isomorphic to $D_4$. My attempt I found that the center $Z(D_4)$ is comprised of $I$ and $R^{2}$, and is normal to $D_4$, so $D_4/Z(D_4)$ is the group ...
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Show $\Bbb{Z}_{30},\ \Bbb{Z}_5 \times D_3,\ \Bbb{Z}_3 \times D_5$ and $D_{15}$ are not isomorphic where $D_n$ is the dihedral group.

I'm looking for a hint for the following problem: Show that no two elements of the following list are isomorphic ($D_n$ is the dihedral group of order $2n$). $$\mathbb{Z}_{30},\ \mathbb{Z}_5 \times ...
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Dihedral group as a subgroup of the symmetric group.

Let $D_n$ be the dihedral group of a regular $n-$polygon. I'd like to write $D_n$ as a subgroup of $S_n$, the set of all permutations of $n$ objects and therefore show why the order of $D_n$ is $2n$. ...
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