NYU Algebra and Rational Numbers Questionnaire

1. Consider Q and Z as additive groups. Let G = Q/Z.
(i) Describe the elements of G.
(ii) Show that every element of G has finite order.
(iii) Is G a finite group? Justify your answer.
2. Determine if O2 (R) is a normal subgroup of GL2 (R).
3. Let O2 (R) be the orthogonal group. Let f, g, h 2 O2 (R) such that f and g are
reflections, and h is a rotation.
(i) Prove that f g is a rotation.
(ii) Prove that f h is a reflection.
(iii) Show that O2 (R) contains an element of order n for every n 2 N.
(iv) Show that O2 (R) contains infinitely many elements of order 2.
4.
(i) Prove that rtv = tu r in E2 , where u = r(v) and r is reflection in the e1 -axis.
(ii) Prove that ⇢✓ tv = tw ⇢✓ in E2 , where w = ⇢✓ (v), and ⇢✓ 2 SO2 (R).
(iii) Let
⇡
, v = (1, 1), x = (1, 0), y = ( 1, 0).
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Illustrate the relation in (ii) by drawing what happens when the isometries
f = ⇢✓ tv and g = tw ⇢✓ are applied to the points x, y 2 R2 .
✓=
(iv) Find the fixed point of the isometry f = ⇢✓ tv when ✓ =
⇡
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5. Let f 2 E2 be a glide reflection. Prove that f 2 is a translation.
6.
(i) Sketch the lattice Z(2, 0) + Z(1, 3).
p
p
(ii) Sketch the lattice Z(1, 3) + Z( 1, 3).
(iii) Sketch the lattice Z(3, 0) + Z(4, 1).
In each case plot at least 20 points.
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and v = (1, 1).
7.
(i) Let
S = {(x, y) 2 Z2 | x 2 Z, y 2 { 1, 1}}
be a frieze pattern in R2 , and let G be the associated frieze group. Find the
translation subgroup of G, and the point group of G.
(ii) Let
T = {(x, y) 2 Z2 | x 2 3Z, y 2 { 2, 2}} [ {(z, 1) 2 Z2 | z ⌘ 1
mod 3}
be a frieze pattern in R2 , and let H be the associated frieze group. Find
the translation subgroup of H, and the point group of H.
(iii) Let
U = {(x, 1) 2 Z2 | x 2 2Z} [ {(z, 1) 2 Z2 | z ⌘ 1
mod 2}
be a frieze pattern in R2 , and let J be the associated frieze group. Find the
translation subgroup of J, and the point group of J.
(iv) Plot the patterns S, T , and U on di↵erent axes, and indicate the reflections,
glide reflections, and rotations in their symmetry groups.
8.
(i) Look around your home or neighborhood for something which has two independent directions of discrete translational symmetry. Take a photo of
the pattern that you find. For example, it could be a piece of fabric, or
kitchen or bathroom tiles, or carpet, or wallpaper, or a fence.
(ii) On the photo, mark the lattice.
(iii) On the photo, indicate all rotations, reflections, and glide reflections. If
there are rotations, what are their orders?
(iv) Compare your pattern with the wallpaper patterns on p174 of Artin (also
posted on Canvas). Which one has the same wallpaper group as your pattern?
(v) Repeat parts (i)–(iv) with a second pattern. Make sure it corresponds to a
di↵erent wallpaper group than your first pattern.
9. Let X = {1, 2, 3, 4} and let the symmetric group S4 act on X by permutations.
(i) What is the orbit of 1 under this action?
(ii) What is the stabilizer of 1 under this action?
(iii) Verify that the orbit-stabilizer theorem holds for this action.
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