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Contents

- Is D8 a cyclic group?
- Is the dihedral group D8 cyclic?
- What is D8 isomorphic to?
- Is D8 a subgroup of S4?
- Is D8 an abelian group?
- What is the center of D8?
- Is D8 isomorphic to S8?
- Is D8 isomorphic to Q8?
- Is dihedral group abelian?
- Which subgroups of D8 are normal?
- Is D8 normal in S4?
- Is D8 isomorphic to D4?
- How many groups of order 8 are there?
- What are the subgroups of D10?
- What are the subgroups of D6?

Thus there are 10 subgroups of D8: the trivial subgroup, the **six cyclic subgroups** {e, s, s2,s3},{e, s2},{e, rx},{e, ry},{e, rx+y}, and {e, rx−y}, the two subgroups {e, s2,rx,ry} and {e, s2,rx+y,rx−y}, and D8. (4b) Show that D8 is not isomorphic to Q8.

The subgroup is (up to isomorphism) cyclic group:Z4 and **the group is (up to isomorphism) dihedral group:D8** (see subgroup structure of dihedral group:D8). The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.

Note that D8 has eight elements. The center of D8 is {R0, R180} (check this). Thus the number of elements in D4/Z(D4) is four, and hence it is isomorphic **either to Z4 or to Z2 ×Z2**.

Solution: (b) D8 is the symmetry group of a square and as such **a Sylow 2-subgroup of S4**.

, which **is abelian**. See center of dihedral group:D8. , which is of prime order, hence its Frattini subgroup is trivial.

Because each automorphism either fixes r or is β composed with an automorphism that fixes r, there are at most 8 automorphisms of D8. Now D8 has a non-trivial center, and in fact its center must have order 2 because G/Z(G) cannot be cyclic unless G is an abelian group. The center of D8 is **the group {1,r2 }**.

Cayley’s theorem says that **D8 is isomorphic to a subgroup of S8**, and gives a homomorphism φ : D8 → S8 where φ(g) is the permutation of the elements of D8 that is given by left- multiplication by g, according to our labeling.

But we now know that D8 has five distinct subgroups of order 2, which shows that **D8 cannot be isomorphic to Q8**.

Small dihedral groups

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

The lattice of subgroups of D8 is given on [p69, Dummit & Foote]. **All order 4 subgroups and 〈r2〉** are normal. Thus all quotient groups of D8 over order 4 normal subgroups are isomorphic to Z2 and D8/〈r2〉 = {1{1,r2},r{1,r2},s{1,r2}, rs{1,r2}} ≃ D4 ≃ V4.

Since the former is 2 and the latter is 3, we see that the order of π(τ) is 1, i.e. τ ∈ kerπ = D8. This means that D8 contains all transpositions and hence is equal to S4, a contradiction. Thus **D8 is not normal in S4**.

These groups are **not isomorphic** to eachother because D8 has an element of order 8 whereas D4 ⊕ Z2 does not.

It turns out that up to isomorphism, there are exactly **5** groups of order 8.

By Lagrange’s Theorem, all the proper subgroups of D10 are cyclic. So the subgroups are: **D10, 〈a〉, 〈b〉, 〈ab〉, 〈a2b〉, 〈a3b〉, 〈a4b〉, {1}**. In particular, there are 8 subgroups. [Diagram of subgroup lattice omitted.]

First, I’ll write down the elements of D6: D6 = {1,x,x2,x3,x4,x5,y,xy,x2y,x3y,x4y,x5y | x6 = 1,y2 = 1,yx = x5y}. This group has order 12, so the possible orders of subgroups are **1, 2, 3, 4, 6, 12**.

- https://www.math.upenn.edu/~chai/370f09/370hwf09/Math370-2.pdf
- https://groupprops.subwiki.org/wiki/Cyclic_maximal_subgroup_of_dihedral_group:D8
- https://www.southalabama.edu/mathstat/personal_pages/jbarnard-archive/teaching/F11-413/Atest3sol.pdf
- http://math.sfsu.edu/beck/435_final.pdf
- https://groupprops.subwiki.org/wiki/Dihedral_group:D8
- https://math.berkeley.edu/~ribet/113/2003/mar20.pdf
- https://math.berkeley.edu/~kpmann/MidtermReview.pdf
- http://home.iiserb.ac.in/~kashyap/MTH%20301/midterm-sol.pdf
- https://en.wikipedia.org/wiki/Dihedral_group
- https://www.math.ucdavis.edu/~wally/teaching/250repository/solutions/homework4_solns.pdf
- http://people.math.binghamton.edu/mazur/teach/40107/40107ex3sol.pdf
- https://mathweb.ucsd.edu/~bprhoades/103Aw14/103AFEReviewSoln.pdf
- https://danaernst.com/the-5-groups-of-order-8/
- http://www.fen.bilkent.edu.tr/~barker/arch323fall14.pdf
- https://sites.math.washington.edu/~palmieri/Courses/2004/Math402/final-sol.pdf

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