Advanced book on Mathematics Olympiad

94 2 Algebra

properties such as optical activity. The point groups of symmetries of molecules were
classified by A. Schönflies as follows:

• Cs:a reflection with respect to a plane, isomorphic toZ 2 ,


• Ci:a reflection with respect to a point, isomorphic toZ 2 ,


• Cn:the rotations by multiples of^2 nπabout an axis, isomorphic toZn,


• Cnv :generated by aCnand aCswith the reflection plane containing the axis of
  rotation; in mathematics this is called the dihedral group,


• Cnh:generated by aCnand aCswith the reflection plane perpendicular to the axis
  of rotation, isomorphic toCn×C 2 ,


• Dn:generated by aCnand aC 2 , with the rotation axes perpendicular to each other,
  isomorphic to the dihedral group,


• Dnd:generated by aCnand aC 2 , together with a reflection across a plane that divides
  the angle between the two rotation axes,


• Dnh:generated by aCnand aC 2 with perpendicular rotation axes, together with a
  reflection with respect to a plane perpendicular to the first rotation axis,


• Sn:improper rotations by multiples of^2 nπ, i.e., the group generated by the element
  that is the composition of the rotation by^2 nπand the reflection with respect to a plane
  perpendicular to the rotation axis,


• Special point groups:C∞v’s andD∞h’s (same asCnvandDnhbut with all rotations
  about the axis allowed), together with the symmetry groups of the five Platonic solids.
  When drawing a molecule, we use the convention that all segments represent bonds
  in the plane of the paper, all bold arrows represent bonds with the tip of the arrow below
  the tail of the arrow. The molecules from Figure 15 have respective symmetry point
  groups the octahedral group andC 3 h.

S

F

F

F

F

F

F

B

O

O O

H

H

H

Figure 15

290.Find the symmetry groups of the molecules depicted in Figure 16.