What is Dihedral: Definition and 42 Discussions

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.
The notation for the dihedral group differs in geometry and abstract algebra. In geometry, Dn or Dihn refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D2n refers to this same dihedral group. The geometric convention is used in this article.

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  1. R

    Show the dihedral group ##D_6## is isomorphic to ##S_3 x Z_2##

    I'm not really sure where to begin with this problem. Any insight as to where to begin or what to look out for would be much appreciated! Orders of ##S_3## ##|e|=1## ##|f|=3## ##|f^2|=2## ##|g|=2## ##|gf|=2## ##|gf^2|=3## Orders of ##Z_2## ##|0|=1## ##|1|=2## Orders of ##S_3 x Z_2##...
  2. C

    MHB Group Ring Integral dihedral group with order 6

    Dear Every one, I am having some difficulties with computing an element in the Integral dihedral group with order 6. Some background information for what is a group ring: A group ring defined as the following from Dummit and Foote: Fix a commutative ring $R$ with identity $1\ne0$ and let...
  3. A

    Isomorphism of dihedral with a semi-direct product

    Homework Statement Let m ≥ 3. Show that $$D_m \cong \mathbb{Z}_m \rtimes_{\varphi} \mathbb{Z}_2 $$ where $$\varphi_{(1+2\mathbb{Z})}(1+m\mathbb{Z}) = (m-1+m\mathbb{Z})$$ Homework Equations I have seen most basic concepts of groups except group actions. Si ideally I should not use them for this...
  4. A

    Sub groups of the dihedral group

    Homework Statement This is only a step in a proof I am trying to make. Let Dm be the dihedral group. r is the rotation of 2π/m around the origin and s is a reflexion about a line passing trough a vertex and the origin. Let<s> and <r> be two subgroups of Dm. Is there a theorem that states...
  5. T

    ##\phi(R_{180})##, if ##\phi:D_n\to D_n## is an automorphism

    Homework Statement Determine ##\phi(R_{180})##, if ##\phi:D_n\to D_n## is an automorphism where ##n## is even so let ##n=2k##. The solutions manual showed that since the center of ##D_n## is ##\{R_0, R_{180}\}## and ##R_{180}## is not the identity then it can only be that...
  6. Mr Davis 97

    1st Isomorphism thm for dihedral gps

    Homework Statement Prove that ##D_\infty/\langle R^n \rangle\cong D_{2n}##, where ##D_\infty=\langle R,S \mid S^2=e, SRS=R^{-1}\rangle##. Homework EquationsThe Attempt at a Solution Pick ##g:\{R,S\} \to D_{2n}## such that ##g(R) = r## and ##g(S) = s##. We note that ##g(S)^2 = 1## and...
  7. R

    Where Can I Find a Reliable Dihedral Angle Database for Computational Chemistry?

    I am working on a computational project and I need to make a series of coordinate system transforms. To do this I will need the dihedral angles for methyl formate, but I cannot find this information anywhere. I've tried the CRC handbook, NIST chemical database, and other data bases, but I can't...
  8. Zafa Pi

    I Can a tetrahedron have all dihedral angles rational?

    At each edge of a tetrahedron the 2 common faces form a dihedral angle. Can each of these 6 angles be rational multiples of pi?
  9. Mr Davis 97

    Showing that dihedral 4 is isomorphic to subgroup of permutations

    Homework Statement D4 acts on the vertices of the square. Labeling them counterclockwise starting from the top left as 1, 2, 3, 4, find the corresponding homomorphism to S4. Homework EquationsThe Attempt at a Solution I am not completely sure what the question is asking. It's pretty clear to...
  10. M

    MHB Subgroups of the dihedral group D6

    Hey! :o I want to make the diagram for the dihedral group $D_6$: Subroups of order $2$ : $\langle \tau \rangle$, $\langle \sigma\tau\rangle$, $\langle\sigma^2\tau\rangle$, $\langle\sigma^3\tau\rangle$, $\langle\sigma^4\tau\rangle$, $\langle\sigma^5\tau\rangle$, $\langle\sigma^3\rangle$...
  11. RJLiberator

    Understanding the Center and Centralizer of a Dihedral Group

    Homework Statement If n ≥ 3, show that Z(D_n) = C(x) ∩ C(y). Homework Equations G is a group, g∈G C(g) = {h∈G: hg = gh } The Centralizer of g Z(G) = {h∈G: hg = gh for all g∈G} The center of G ∩ means the set of all points that fall in C(x) and C(y). Every element of D_n can be uniquely...
  12. Avatrin

    Subgroups of Symmetric and Dihedral groups

    I am having problem working with the objects in the title. Working with permutations, rotations and reflections is fine, but I have problem with the following: Showing a subgroup is or is not normal (usually worse in the case of symmetric groups) Finding a subgroup of order n. Showing that...
  13. T

    Quick question about subgroups of "odd" dihedral groups

    Will odd dihedral groups (e.g. d6, d10, d14) only have the identity, σ, τ and itself as subgroups as any power of σ with τ generates the entire group? If so would the subgroup lattice of d14 just be: d14 → σ & τ → e Thanks!
  14. B

    Dihedral Group D_4: Denotations & Correspondence

    In my class, we have denoted the elements of the dihedral group ##D_4## as ##\{R_0, R_{90}, R_{180}, R_{270}, F_{\nearrow}, F_{\nwarrow}, F_{\leftrightarrow}, F_{\updownarrow} \}##. Not surprising, I was rather bewildered when I searched the internet for information on this group and encountered...
  15. B

    Proving that the Dihedral Groups are non-cylic

    Hello everyone, I am suppose to show that all the Dihedral groups (##D_n##, for ##n >2##) are noncyclic. I know that every cyclic group must be abelian. So, what I intended on showing was that at least two elements in ##D_n## are not commutative. Here are my thoughts: Because we are dealing...
  16. Greg Bernhardt

    What Are the Properties of Dihedral Groups?

    Definition/Summary The dihedral group D(n) / Dih(n) is a nonabelian group with order 2n that is related to the cyclic group Z(n). The group of symmetries of a regular n-sided polygon under rotation and reflection is a realization of it. The pure rotations form the cyclic group Z(n), while...
  17. O

    MHB Using dihedral group in Lagrange theorem

    i was given that D4=[e,c,c2,c3,d,cd,c2d,c3d] therfore D4=<c,d> is the subgroup of itself generated by c,d then they defined properties of D4 as follows ord(c)=d, ord(d)=2, dc=c-1d i am strugging to understand how they got that c4=e=d2
  18. U

    Finding a third plane that has a dihedral angle to two other planes.

    Homework Statement The acute angle between two planes is called the dihedral angle. Plane x−3y+2z=0 and plane 3x−2y−z+3=0 intersect in a line and form a dihedral angle θ . Find a third plane (in point-normal, i.e. component, form) through the point (-6/7,0,3/7) that has dihedral angle θ/2 with...
  19. srfriggen

    Quick definition question: Dihedral group

    A dihedral group of an n-gon denoted by Dn, whose corresponding group is called the Dihedral group of order 2n? What I gather from that is a square has 8 symmetries, an octagon has 16, a hexagon 12, etc?
  20. R

    How would one prove the dihedral group D_n is a group?

    I don't understand how to show that the reflections and rotations are associative. Thanks for any help.
  21. T

    Dihedral Group on a square

    Homework Statement Let G=D_4 (the group of symmetries (reflections/rotations) of a square) and let X=\{ \text{colourings of the edges of a square using the colours red or blue} \} so a typical element of X is: What is the size of X? Let G act on X in the obvious way. You are given...
  22. B

    Isomorphism of the Dihedral group

    We're doing isomorphisms and I was just wondering, is the dihedral group D_{12} isomorphic to the group of even permutations A_4?
  23. F

    Is N3 a Subgroup of Dihedral Group Dih(12)?

    Homework Statement Taking the Dih(12) = {α,β :α6 = 1, β2 = 1, βα = α-1β} and a function Nr = {gr: g element of Dih(12)} Homework Equations Taking the above I have to find the elements of N3. And then prove that N3 is not a subgroup of Dih(12). The Attempt at a Solution For N3 I...
  24. S

    Subgroups of dihedral group and determining if normal

    To find all subgroups you use the fact that by Legrange theorem and subgroup will divide the order of the group, so for the dihedral group D4 our subgroups are of order 1,2, and 4. I am unsure how to tell whether or not these groups will be normal or not.
  25. S

    Prove reflections generate a the dihedral group Dn

    Homework Statement Let l_{1} and l_{2} be the lines through the origin in \Re^{2} that intersect in an angle \pi/n and let r_{i} be the reflection about l_{i}. Prove the r_{1} and r_{2} generate a dihedral group D_{n}. Homework Equations D_{n}: the dihedral group of order 2n generated by...
  26. C

    Abstract Algebra dihedral group

    Homework Statement 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~=D_2n, where |xy|=n. I have no idea how to solve this or even where to begin. I tried setting up G=<x,y|x^2=y^2=1=(xy)^n> But couldn't get any farther, I am so...
  27. M

    DIHEDRAL GROUP - Internal Direct Product

    I have to prove that D4 cannot be the internal direct product of two of its proper subgroups.Please help.
  28. M

    Why can't Dn be isomorphic to the direct product of its subgroups?

    The dihedral group Dn of order 2n has a subgroup of rotations of order n and a subgroup of order 2. Explain why Dn cannot be isomorphic to the external direct product of two such groups. Please suggest how to go about it. If H denotes the subgroup of rotations and G denotes the subgroup of...
  29. B

    Show that the matrix representation of the dihedral group D4 by M is irreducible.

    1. Homework Statement [/b] Show that the matrix representation of the dihedral group D4 by M is irreducible. You are given that all of the elements of a matrix group M can be generated from the following two elements, A= |0 -1| |1 0| B= |1 0| |0 -1| in the sense that all...
  30. S

    Dihedral angles for expanded icosidodecahedron.

    I am trying to determine two dihedral angles for this polyhedron: http://georgehart.com/virtual-polyhedra/vrml/zonish-6-icosidodecahedron.wrl 1. The angle between a pentagon and a square? 2. The angle between a square and a triangle? This polyhedron is formed by adding squares...
  31. M

    Proving Non-Cyclic and Non-Abelian Properties of Dihedral and Symmetric Groups?

    Homework Statement Prove that S_n and D_n for n>=3 are non-cyclic and non-abelian. Homework Equations I get that I need to show that two elements from each group do not commute and that there is not a single generator to produce the groups... I am just unsure of how to do this...
  32. A

    Problem regarding Dihedral of wigs

    hi. i have a doubt. Which configuration is more advantageous to the wings in terms of lift generated and stability? 1. a wing with a full span dihedral from fuselage to the tip 2. A wing with a partial span range dihedral from tips to somewhere in the wings and then straight till fuselage 3. a...
  33. I

    Dihedral group D5 - Symmetry of a Pentagon - Conjugacy classes

    Hi I am struggling to get my head fully around the conjugacy classes of D5. Everywhere I have looked seems to say that there are 4 irreducible representations of D5 which implies that there are 4 conjugacy classes. However, when examining the symmetry of the pentagon I am only able to see 3...
  34. T

    What are the subgroups of D12 and how can they be proven to be isomorphic to Dm?

    Homework Statement 1. Let Dn be the dihedral group of order 2n, n>2 . A. Prove that each non-commutative sub-group of Dn isomorphic to Dm for some m. B. Who are all the non-commutative subgroups of D12? 2. Let G be the group of all the matrices from the form: 1 a c 0 1 b 0 0...
  35. B

    Isomorphism between Dihedral and Symmetric groups of the same order?

    Is there a way to prove generally that the Dihedral group and its corresponding Symmetric group of the same order are isormorphic. In class we were only shown a particular example, D3 (or D6 whatever you wish to use) and S3, and a contructed homomorphism, but how could you do it generally? Would...
  36. K

    Proof on Dihedral Groups

    Let n be a positive integer and let m be a factor of 2n. Show that Dn (the dihedral group) contains a subgroup of order m. I'm not really sure where to start with this one. I know that Dn is generated by two types of rotations: flipping the n-gon over about an axis, and rotating it 2π/n...
  37. S

    Regarding signed dihedral angle doubt

    Hi to all.. I am doing Phd in bioinformatics, having little knowledge abt maths.. I have calculated Phi Psi angles(dihedral angle)between two planes... But the thing is i didnt get proper sign for my result(may be + or -). I used following formulae for calculate the dihedral angle between...
  38. N

    Groups, Normalizer, Abstract Algebra, Dihedral Groups help?

    [b]1. Let G be a Group, and let H be a subgroup of G. Define the normalizer of H in G to be the set NG(H)= the set of g in G such that gHg-1=H. a) Prove Ng(H) is a subgroup of G b) In each of the part (i) to (ii) show that the specified group G and subgroup H of G, CG(H)=H, and NG(H)=G...
  39. J

    Total and dihedral energies ffrom MD simulations

    Hi guys, I hope you can give me any idea about: After a long molecular dynamic simulation of a polymer under periodic boundary conditions, dihedral and total energies are lower than zero, (negative). do you know the physical meaning of that result? thanks for reading and for your help...
  40. M

    A_n Group: Dihedral Group D_n?

    My question is quite basic... sorry. What is the A_n group? Is it the dihedral group (represented in my book by D_n)? Thanks
  41. S

    Geometric Significance of the Dihedral Group D/o

    Consider the dihedral group D/o, generated by x and y where o(x)=2 and o(y)=5 What is the geometric significance of D/o? Which of G/<x> and G/<y> are well defined groups? Give reasons?
  42. W

    Proving g2 = u in Dihedral Group of Order 8

    Hello, I am having trouble understanding groups in my group theory class. I am not confident on how to approach the following question: I know that y4 = u. So then, g = xy4 = xu = x. Then g2 = x2 = u which is what I am trying to prove. Now if i = 1 then, g = xy. Then...
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