1. Properties of point groups:
If two symmetry operations a,b  G, then consecutive execution of these operations is also an element of G (multiplication): a,b  G  a x b  G Symmetry operations are executed from the right (b first, then a) Example: C2 x v = v’
The result of consecutive symmetry operations does not depend on how they are combined: a,b,c  G  (a x b) x c = a x (b x c) Note: This does not mean that the order can change!
There exists a neutral element E  G for which a x E = E x a for all a  G (Example: I)
For each element a  G there exists an inverse element a-1  G for which a x a-1 = a-1 x a = E (Example: v-1 = v)

2. Note:
If for all elements a,b  G a x b = b x a, then the group is called commutative group or “Abelian group”.
The number N of the elements of a group is called the order of the group
If the element a  G, then all powers of a are  G. There is a finite number p < N with ap = E
Two elements a,b  G belong to the same class if there is an element c  G so that a = c-1 x b x c No element can belong to two classes of a group. Example: clockwise C3 rotation and counterclockwise C3 rotation: a = C3; b = C32; c = c-1 = v
Multiplication tables show the results of all products between symmetry operations of a point group

3. from W. Demtröder “Molecular Physics”

4. Molecular point groups: The non-axial groups C1, Cs, Ci
C1 only has a “symmetry axis” for rotation by 360º = E Example: CFClIBr
Cs has E and a symmetry plane labeled h by convention Example: CH3NO2
Ci only has E and i Example: 1,2-dichloro-1,2-difluoromethane
C1 point group: CFClBrI
CS point group: CH3NO2
Ci point group: 1,2-dichloro-1,2-difluoroethane

5. Molecular point groups: The cyclic groups Cn, Sn
Cn contains E and Cn axes Example: H2O2 (C2)
Sn contains E and Sn axes. For S2n, there is also a Cn axis. Example: 12-crown-4 (S4)
C2 point group: H2O2
S4 point group: 12-crown-4

6. Molecular point groups: Axial groups with mirror planes Cnh, Cnv
contain one Cn axis
Cnh contains 1 horizontal plane h, inversion
Example: butadiene C2h
Cnv contains n vertical planes v Example: CH3Cl (C3v); water (C2v)
C2h point group: N2F2
C2h point group: butadiene

7. Axial groups with multiple rotation axes Dn, Dnd, Dnh
Linear molecules: Cv, Dh
Cubic groups: Td and Oh
Icosahedral group Ih

8. from W. Demtröder “Molecular Physics”