CHAPTER 6: Question 7

Simple Answers

a)     -x, -y, z
b)     -x, y, -z
c)     x, -y, -z
d)     x, y, -z
e)     x, -y, z
f)     -x, y, z

S₂¹ = C₂¹ (about the z axis) then s (in the x,y plane)

consider a point x,y,z
C₂¹(z) transforms this to -x, -y, z
s(xy) then transforms this point to -x, -y, -z

This is the same as the effect that i would have on the point. Therefore, S₂¹ = i

S₄¹ = C₄¹ (about the z axis) then s (in the x,y plane)

consider a point x,y,z
C₄¹(z) transforms this to x, -y, z
s(xy) then transforms this to x, -y, -z

To obtain the effect of S₄², simply apply C₄¹(z) and s(xy) again starting from x, -y, -z

C₄¹(z) transforms this to -x, -y, -z
s(xy) then transforms this to -x, -y, z

This is the same effect that C₂¹ would have on the point. Therefore S₄² = C₂¹