# 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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### 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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### 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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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: $$D_{\infty} = \; \left<r,s \;|\; srs=r^... -1 votes 2 answers 112 views ### Prove that the groups D_6 and \mathbb{Z}_6 are isomorphic I'm stuck trying to prove that: Prove that the groups D_6 and \mathbb{Z}_6 are isomorphic. My attempt: Let D_6=\{e,a,a^2,b,ab,a^2b\} where a^3=b^2=e and ba=a^2b and \mathbb{Z}_6=\{,,[... 3 votes 1 answer 95 views ### Each extra-special group of order 2^{2n+1} is a central product of D_8s or of D_8s and a single Q_8. This is Exercise 5.3.7(i) of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to this search, it is new to MSE. This is a classification problem. This Wikipedia ... 1 vote 1 answer 80 views ### Show that D_4 is isomorphic to the direct product of two cyclic groups of order 2. What I have tried is the following: Since D_4 is a group of order four, then it is abelian, so we use the fundamental theorem of abelian groups which tells us that every finite abelian group is ... 4 votes 2 answers 191 views ### Showing {\rm Aut}(D_{2^n})\cong{\rm Aut}(Q_{2^n}) for n\ge 4. In this comment from back in 2013, it is claimed that$${\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 ... 0 votes 0 answers 46 views ### 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 ... 1 vote 1 answer 49 views ### 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 ... 3 votes 0 answers 56 views ### 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 ... 1 vote 0 answers 22 views ### 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 ... 1 vote 1 answer 56 views ### 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), ... 0 votes 0 answers 13 views ### 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 ... 1 vote 1 answer 65 views ### 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(\... 2 votes 1 answer 26 views ### 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 ... 2 votes 0 answers 91 views ### 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{... 1 vote 0 answers 40 views ### 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. ... 1 vote 0 answers 53 views ### 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 ... 1 vote 1 answer 97 views ### 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 ... -2 votes 1 answer 86 views ### Accessing Elements of Group in GAP [closed] ... 1 vote 1 answer 95 views ### 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  |\... 2 votes 1 answer 65 views ### 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-... 0 votes 0 answers 85 views ### 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 ... 0 votes 1 answer 70 views ### 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 ... 1 vote 0 answers 28 views ### 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 ... 1 vote 3 answers 160 views ### 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 (... 1 vote 2 answers 77 views ### 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 ... 0 votes 0 answers 43 views ### 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 ... 1 vote 1 answer 43 views ### 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 ... 3 votes 1 answer 57 views ### 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 ...
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 $2n$. ...