Group Theory is a mathematical method by which aspects of a molecules symmetry can be determined.
The symmetry of a molecule reveals information about its properties (i.e., structure, spectra, polarity, chirality, etc…)
Clearly, the symmetry of the linear molecule ABA is different from AAB.
In ABA the AB bonds are equivalent, but in AAB they are not.
However, important aspects of the symmetry of H_{2}O and CF_{2}Cl_{2} are the same. This is not obvious without Group Theory.
A molecule or object is said to possess a particular operation if that operation when applied leaves the molecule unchanged.
Each operation is performed relative to a point, line, or plane  called a symmetry element.
There are 5 kinds of operations
1. Identity
2. nFold Rotations
3. Reflection
4. Inversion
5. Improper nFold Rotation
does nothing, has no effect
all molecules/objects possess the identity operation, i.e., posses E.
E has the same importance as the number 1 does in multiplication (E is needed in order to define inverses).
rotation by 360°/n about a particular axis defined as the nfold rotation axis.
C_{2} = 180° rotation, C_{3} = 120° rotation, C_{4} = 90° rotation, C_{5} = 72° rotation, C_{6} = 60° rotation, etc.
Rotation of H_{2}O about the axis shown by 180° (C_{2}) gives the same molecule back.
Therefore H_{2}O possess the C_{2} symmetry element.
However, rotation by 90° about the same axis does not give back the identical molecule
Therefore H_{2}O does NOT possess a C_{4} symmetry axis.
BF_{3} posses a C_{3} rotation axis of symmetry.
(Both directions of rotation must be considered)
This triangle does not posses a C_{3} rotation axis of symmetry.
XeF_{4} is square planar.
It has four DIFFERENT C_{2} axes
It also has a C_{4} axis coming out of the page called the principle axis because it has the largest n.
By convention, the principle axis is in the zdirection
If reflection about a mirror plane gives the same molecule/object back than there is a plane of symmetry (s).
If plane contains the principle rotation axis (i.e., parallel), it is a vertical plane (s_{v})
If plane is perpendicular to the principle rotation axis, it is a horizontal plane (s_{h})
If plane is parallel to the principle rotation axis, but bisects angle between 2 C_{2} axes, it is a diagonal plane (s_{d})
H_{2}O posses 2 s_{v} mirror planes of symmetry because they are both parallel to the principle rotation axis (C_{2})
XeF_{4} has two planes of symmetry parallel to the principle rotation axis: s_{v}
XeF_{4} has two planes of symmetry parallel to the principle rotation axis and bisecting the angle between 2 C_{2} axes : s_{d}
XeF_{4} has one plane of symmetry perpendicular to the principle rotation axis: s_{h}
The operation is to move every atom in the molecule in a straight line through the inversion center to the opposite side of the molecule.
Therefore XeF_{4} posses an inversion center at the Xe atom.
nfold rotation followed by reflection through mirror plane perpendicular to rotation axis
Note: n is always 3 or larger because S_{1} = s and S_{2} = i.
These are different, therefore this molecule
does not posses a C_{3} symmetry axis.
This molecule posses the following symmetry elements: C_{3}, 3 s_{d}, i, 3 ^ C_{2}, S_{6}. There is no C_{3} or s_{h}.
Eclipsed ethane posses the following symmetry elements: C_{3}, 3 s_{v}, 3 ^ C_{2}, S_{3}, s_{h}. There is no S_{6} or i.
Compiling all the symmetry elements for staggered ethane yields a Symmetry Group called D_{3d}.
Compiling all the symmetry elements for eclipsed ethane yields a Symmetry Group called D_{3h}.
Symmetry group designations will be discussed in detail shortly
1. Any result of two or more operations must produce the same result as application of one operation within the group.
i.e., the group multiplication table must be closed
Consider H_{2}O which has E, C_{2} and 2 s_{v}'s.
i.e.,
of course
etc…
The group multiplication table obtained is therefore:
E 
C_{2} 
s_{v} 
s'_{v} 

E 
E 
C_{2} 
s_{v} 
s'_{v} 
C_{2} 
C_{2} 
E 
s'_{v} 
s_{v} 
s_{v} 
s_{v} 
s'_{v} 
E 
C_{2} 
s'_{v} 
s'_{v} 
s_{v} 
C_{2} 
E 
Note: the table is closed, i.e., the results of two operations is an operation in the group.
2. Must have an identity ()
3. All elements must have an inverse
i.e., for a given operation () there must exist an operation () such that
Certain symmetry operations can be present simultaneously, while others cannot.
There are certain combinations of symmetry operations which can occur together.
Symmetry Groups combine symmetry operations that can occur together.
Symmetry groups contain elements and there mathematical operations.
For example, one of the symmetry element of H_{2}O is a C_{2}axis. The corresponding operation is rotation of the molecule by 180° about an axis.
C_{1}: only E 
C_{s}: E and s only 
C_{i}: E and i only 
C_{n}: E and C_{n} only 
C_{2:} 
C_{3:} 
C_{nv}: E and C_{n} and n s_{v}'s 
C_{2v}: E, C_{2}, 2 s_{v} H_{2}O C_{3v}: E, C_{3}, 3 s_{v} NH_{3} C• _{v}: E, C• , • s_{v} HF, HCN 
C_{nh}: E and C_{n} and s_{h} (and others as well) 
C_{2h}: E, C_{2}, s_{h}, i 
D_{n}: E, C_{n}, n C_{2} axes ^ to C_{n} 
D_{3}: E, C_{3}, 3 ^ C_{2} [Co(en)_{3}]^{3+} 
D_{nh}: E, C_{n}, n C_{2} axes ^ to C_{n}, s_{h} 
D_{3h}: E, C_{3}, 3 ^ C_{2}, s_{h} 
D_{3h}: E, C_{3}, 3 ^ C_{2}, s_{h} eclipsed ethane 
D_{6h}: E, C_{6}, 6 ^ C_{2}, s_{h} 
D• _{h}: E, C• , • ^ C_{2}, s_{h} H_{2} 
D_{nd}: E, C_{n}, n C_{2} axes ^ to C_{n}, 
D_{3d}: E, C_{3}, 3 ^ C_{2}, 3 s_{d} staggered ethane 
S_{2n}: E, C_{n}, S_{2n} (no mirror planes) S_{4}, S_{6}, S_{8}, etc. (Note: never S_{3}, S_{5}, etc.) 
S_{4}: E, C_{2}, S_{4} 
T_{d}: E, 8 C_{3}, 3 C_{2}, 6 S_{4}, 6 s_{d} Tetrahedral structures No need to identify all the symmetry elements  simply recognize T_{d} shape. methane, CH_{4} 
O_{h}: E, 8 C_{3}, 6 C_{2}, 6 C_{4}, i, 6 S_{4}, 8 S_{6}, 3 s_{h}, 6 s_{d} Octahedral structures No need to identify all the symmetry elements  simply recognize O_{h} shape. 
I_{h}: E, 12 C_{5}, 20 C_{3}, 15 C_{2}, i, 12 S_{10}, 20 S_{6}, 15 s Icosahedron 
Other rare high symmetry groups are T, T_{h}, O, and I 