The dihedral group D4 is **the symmetry group of the square**: let S=ABCD be a square. The different symmetry maps of S are: the identity map e. the rotations r,r2,r3 of 90∘,180∘,270∘ around the center of S counterclockwise.

Contents

- What is the order of dihedral group D4?
- What elements are in dihedral D4?
- What is dihedral group Dn?
- What is center D4?
- What is D4 isomorphic to?
- Is D4 an Abelian group?
- How many elements does D4 have?
- What is dihedral group D5?
- Is dihedral group abelian?
- What is dihedral group d8?
- What is dihedral group D1?
- What is dihedral group D2?
- What are the subgroups of D4?
- Is D4 isomorphic to Q8?
- What is the center of DN?

B = (0 1 1 0 ) is a D4 group. Theorem 1 (Properties of D4 .). If G is a D4 group, then G is a non-commutative group of **order 8**, where each element of D4 has the form aibj,0 ≤ i ≤ 3.0 ≤ j ≤ 1. It is now clear that the group D4 is unique up to isomorphism.

The group D4 has **eight elements, four rotational symmetries and four reflection symmetries**. The rotations are 0◦, 90◦, 180◦, and 270◦, and the reflections are defined along the four axes shown in Figure 1. We denote these elements as σ0, σ1, …, σ7.

The dihedral group Dn is **the symmetry group of a regular polygon with n vertices**. We imagine this polygon to have vertices on the unit circle, where the vertices are labeled 0,1,…,n−1, starting at (1,0) and proceeding counterclockwise at angles in multiples of 360 /n degrees, i.e. H. 2π/n radians.

Center of the dihedral group D4

D4=⟨a,b:a4=b2=e,ab=ba−1⟩ The center of D4 is given by: **Z(D4)= {e ,a2}**

Therefore the number of elements in D4/Z(D4) is four, and therefore it is isomorphic to either **Z4 or Z2 ×Z2**.

We see that **D4 is not abelian**; the Cayley table of an abelian group would be symmetric over the main diagonal.

Thus D4 has **a 2-element normal subgroup and three 4-element subgroups**.

The dihedral group D5 is **the symmetry group of the regular pentagon**: let P=ABCDE be a regular pentagon. The different symmetry maps of P are: the identity map e. the rotations r,r2,r3,r4 of 72∘,144∘,216∘,288∘ around the center of P counterclockwise.

Small dihedral groups

**D _{1} and D_{2} are the only abelian dihedral groups**.

The dihedral group , sometimes called , also called the eighth-order dihedral group or the four-element dihedral group, is defined by the following representation: **The row element is multiplied on the left and the column element is multiplied on the right**. item.

The dihedral group D1 is **the symmetry group of the segment**: Let AB be a segment. The symmetry maps of AB are: The identity map e. The rotation r of 180∘ around the center of AB.

The dihedral group D2 is **the symmetry group of the rectangle**: Let R=ABCD be a (non-square) rectangle. The different symmetry maps of R are: The identity map e. The rotation r (in both directions) of 180∘

(a) The real normal subgroups of D4 = {e, r, r2,r3, s, rs, r2s, r3s} are {e, r, r2,r3}, {e, r2, s, r2s} , {e, r2, rs, r3s} and {e, r2}.

The groups D4 and Q8 are **not isomorphic** because there are 5 elements of order 2 in D4 and only one element of order 2 in Q8.

By definition, the center of Dn is: **Z(Dn)={g∈Dn:gx=xg,∀x∈Dn}**

- https://wiki.ubc.ca/images/f/fe/Dihedral.pdf
- https://mdpi-res.com/d_attachment/symmetry/symmetry-12-00548/article_deploy/symmetry-12-00548.pdf
- https://web.northeastern.edu/suciu/MATH3175/DihedralGroups.pdf
- https://proofwiki.org/wiki/Dihedral_Group_D4/Center
- https://www.southalabama.edu/mathstat/personal_pages/jbarnard-archive/teaching/F11-413/Atest3sol.pdf
- http://users.jyu.fi/~laurikah/REP/REPtext2010_1.pdf
- https://math.berkeley.edu/~serganov/114/gsolhw.pdf
- https://proofwiki.org/wiki/Definition:Dihedral_Group_D5
- https://en.wikipedia.org/wiki/Dihedral_group
- https://groupprops.subwiki.org/wiki/Element_structure_of_dihedral_group:D8
- https://proofwiki.org/wiki/Definition:Dihedral_Group
- https://proofwiki.org/wiki/Dihedral_Group/Examples/D2
- http://pi.math.cornell.edu/~gomez/Math3560/Files/3560hw8sol.pdf
- https://kconrad.math.uconn.edu/blurbs/grouptheory/groupsp3.pdf
- https://proofwiki.org/wiki/Center_of_Dihedral_Group

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