MAT 150A, Fall 2015
Solutions to Homework Assignment 6
Page 188 3.2. Let m be an orientation-reversing isometry. Prove algebraically that
m is a translation.
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Solution: By classification theorem, every orientation-reversing isometry is a glide
translation, that is, reflection r in some line l composed with the translation t by some
vector v parallel to l: m = tr. Since the reflection in l commutes with the translation, we
have that m2 = trtr = t2 r2 = t2 , so it is a translation by 2v.
5.1. Let l1 and l2 be lines through the origin in R2 that intersect at angle π/n, and let
ri be the reflection about li . Prove that r1 and r2 generate a dihedral group Dn .
Solution: A composition of these reflections is a rotation by 2π/n, and all other
elements in Dn are either powers of this rotation or compositions of either of ri with a
power of this rotation. Indeed, a composition of a rotation by an angle φ with a reflection
is another reflection such that that the angle between the reflecting lines equals φ/2.
5.5. Prove that the group of symmetries of the frieze pattern CCCCCCC is isomorphic
to Z2 × Z.
Solution: Let s be a reflection in the x-axis, and let t be a translation to the right
by 1 unit. Then st = ts is the glide reflection, t2 is a translation by 2 units etc. Every
isometry has the form sa tb (since st = ts), and since s2 = e we get a = 0 or a = 1. We
can identify this with Z2 × Z by writing (0, k) = tk and (1, k) = stk .
Page 190 7.5. Let G be the group of symmetries of a cube, including the orientationreversing symmetries. Describe the elements of G geometrically.
Solution: The groups of symmetries of a cube has 48 elements:
(1) identity I
(2) central symmetry −I
(3) 6 rotations by 90◦ around axis perpendicular to the faces (3 axis, 2 rotations per
each)
(4) 6 same rotations composed with reflections in perpendicular planes
(5) 3 rotations by 180◦ around the same axis
(6) 3 same rotations composed with reflections
(7) 8 rotations by 120◦ around axis through the opposite vertices of the cube (4 axis,
2 rotations per each)
(8) 8 same rotations composed with reflections
(9) 6 rotations by 180◦ around the axis connecting the centres of opposite edges
(10) 6 same rotations composed with reflections
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