First Course, 2nd. ed.

A Solutions Manual fo Abstract

Algebra:

A First Course, 2nd. ed.Author Stephen Lovett All Chapters 1 / 4

1|Groups Hints and Corrections ˆExercise 1.7.14 is simply wrong.

1.1 – Symmetries of a Regular Polygon

Exercise:1 Section 1.1

Question:Use diagrams to describe all the dihedral symmetries of the equilateral triangle.

Solution:The equilateral triangle has 6 dihedral symmetries.

identityrotation 120 ◦ rotation 240 ◦ reflection throughx-axisreflectionreflection

Exercise:2 Section 1.1

Question:Write down the composition table forD4.

Solution:Composition table forD4where the entries givea◦b.

a\b1 r r 2 r 3 s sr sr 2 sr 3

• 1 r r

2 r 3 s sr sr 2 sr 3 r r r 2 r 3 1sr 3 s sr sr 2 r 2 r 2 r 3

• r sr

2 sr 3 s sr r 3 r 3

• r r

2 sr sr 2 sr 3 s s s sr sr 2 sr 3

• r r

2 r 3 srsr sr 2 sr 3 s r 3

• r r

2 sr 2 sr 2 sr 3 s sr r 2 r 3

• r

sr 3 sr 3 s sr sr 2 r r 2 r 3 1 (1.1)

Exercise:3 Section 1.1

Question:Determine whatr

3 sr 4 srcorresponds to in dihedral symmetry ofD8.Solution:In dihedral symmetry ofD8, we have the following algebraic identities onrands: r 8 = 1, s 2 = 1, r k s=sr −k .

So for our element, progressively change it to put all thesterms to the left:

r 3 sr 4 sr=r 3 s(r 4 s)r=r 3 s 2 r −4 r=r 3 1r −3 = 1.

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4CHAPTER 1. GROUPS

Exercise:4 Section 1.1

Question:Determine whatsr

6 sr 5 srscorresponds to as a dihedral symmetry ofD9.

Solution:Recall from Corollary 3.5 thatsr

k =r n−k swhere in our casen= 9. So, sr 6 sr 5 srs=sr 6 sr 5 ssr 8 =ssr 3 r 5 (1)r 8 = (1)r 8 r 8 =r 9 r 7 = 1r 7 =r 7 .

Exercise:5 Section 1.1

Question:Letnbe an even integer withn≥4. Prove that inDn, the elementr

n/2 satisfiesr n/2 w=wr n/2 for allw∈Dn.

Solution:From the paragraph above Proposition 3.1.4 we can writew∈Dnasw=s

a r b whereais either 0 or 1. Considerr n/2 s a r b . We have two cases.

Case 1:a= 0r

n/2 r b =r n/2+b =r b+n/2 =r b r n/2 .

Case 2:a= 1 r

n/2 sr b =sr n−n/2 r b =sr n/2 r b =sr n/2+b =sr b+n/2 =sr b r n/2 .In both cases, we see thatr n/2 w=wr n/2 .

Exercise:6 Section 1.1

Question:Letnbe an arbitrary integern≥3. Show that an expression of the form

r a s b r c s d

· · ·

is a rotation if and only if the sum of the powers onsis even.

Solution:For any numberslandmwe haver

l s m =s m r l−m . So we can move all powers ofsaround without changing the exponent’s value. Since we can rewrite any element ass j r k , we haver a s b r c s d · · ·=s b+d+··· r m for somem. Now, ifb+d+· · ·is an even number thens b+d+··· r m =s 2 s 2 · · ·s 2 r m = (1)(1)· · ·(1)r m = 1r m =r m and our element is a rotation. Ifb+d+· · ·is an odd number thens b+d+··· r m =s 1 s 2 · · ·s 2 r m =s(1)(1)· · ·(1)r m =sr m and our elements is not a rotation.

Exercise:7 Section 1.1

Question:Use linear algebra to prove that

Rα◦Fβ=F

α/2+β, Fα◦Rβ=F

α−β/2,andFα◦Fβ=R

2(α−β).

Solution:As linear transformations onR

2 →R 2 , the matrices of the rotationRαand of the reflectionFβwith respect to the standard basis are respectively ⊆ cosα−sinα sinαcosα ⊇ and ⊆ cos 2βsin 2β sin 2β−cos 2β ⊇ .The matrix forRα◦Fβis ⊆ cosα−sinα sinαcosα

⊇ ⊆

cos 2βsin 2β sin 2β−cos 2β ⊇ = ⊆ cosαcos 2β−sinαsin 2βcosαsin 2β+ sinαcos 2β sinαcos 2β+ cosαsin 2βsinαsin 2β−cosαcos 2β ⊇ = ⊆ cos(α+ 2β) sin(α+ 2β) sin(α+ 2β)−cos(α+ 2β) ⊇

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1.1. SYMMETRIES OF A REGULAR POLYGON5

This matrix corresponds to the reflectionF

α/2+β.

The matrix forFα◦Rβis ⊆ cos 2αsin 2α sin 2α−cos 2α

⊇ ⊆

cosβ−sinβ sinβcosβ ⊇ = ⊆ cos 2αcosβ+ sin 2αsinβ−cos 2αsinβ+ sin 2αcosβ sin 2αcosβ−cos 2αsinβ−sin 2αsinβ−cos 2αcosβ ⊇ = ⊆ cos(2α−β) sin(2α−β) sin(2α−β)−cos(2α−β) ⊇ This matrix corresponds to the reflectionF

α−β/2.

The matrix forFα◦Fβis ⊆ cos 2αsin 2α sin 2α−cos 2α

⊇ ⊆

cos 2βsin 2β sin 2β−cos 2β ⊇ = ⊆ cos 2αcos 2β+ sin 2αsin 2βcos 2αsin 2β−sin 2αcos 2β sin 2αcos 2β−cos 2αsin 2βsin 2αsin 2β+ cos 2αcos 2β ⊇ = ⊆ cos(2α−2β)−sin(2α−2β) sin(2α−2β) cos(2α−2β) ⊇ This matrix corresponds to the reflectionR

2(α−β).

Exercise:8 Section 1.1

Question:Describe the symmetries of an ellipse with unequal half-axes.

Solution:The ellipse with unequal half-axes has 4 symmetries. Supposing that the axes of the ellipse are on thexandyaxes, then the ellipse has for symmetries: the identity, reflection through thexaxis, reflection through theyaxis, and rotation by 180 ◦ , which is the composition of the two reflections.

Exercise:9 Section 1.1

Question:Determine the set of symmetries for each of the following shapes (ignoring shading): (a)(b)(c)(d)(e)(f)

Solution:

a) b)D3.c)D12.d) e)D5.f) ◦ degree rotational symmetry.

Exercise:10 Section 1.1

Question:Sketch a pattern/shape (possibly a commonly known logo) that hasD8symmetry but does not haveDnsymmetry forn >8.Solution:Here is an example ofD8symmetry:

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