A point group consists of geometric transformations known as symmetry operations, which preserve a single common point while transforming an object defined in a real vector space into physically indistinguishable replicas of itself.
Although an object undergoing a symmetry operation ends up looking the same after the transformation, the labelling of similar components of the object may change. In other words, to form a point group, all symmetry operations for an object must:
-
- Send the object into physically indistinguishable copies of itself.
- Combine with one another through binary operations such that the results are consistent with the 4 properties of a group.
- Leave one point invariant.
Point groups are determined by considering symmetry operations for different objects, beginning with simple shapes and moving on to more complex ones. According to the three abovementioned requirements, we start by inspecting the chosen object visually and finding all the symmetry elements (not to be confused with group elements) and their associated symmetry operations. For example, the only symmetry elements for the object in figure I, which is made up of two equally spaced right-angled triangles on a circle, are the identity symmetry element 
and a 2-fold rotation axis 
. The corresponding symmetry operations are 
and 
.
Next, we select a position vector 
(see figure II) and perform the symmetry operations and consecutively on the position vector. The results, in relation to the transformation of the position vector, are summarised in the multiplication table i:
Note that we have omitted the carets – i.e. 
– for simplicity. From the table, we can easily verify all four properties of the group, e.g. the identity element is and the inverse of an element of the group is the element itself. Therefore, the set of symmetry operations 
for the object forms a point group of order 2 under the binary operation of multiplication. We call this group the point group. Similarly, the set of symmetry operations 
for the object in figure III forms the point group 
of order 3 (see multiplication table ii). In general, we have an infinite number of uniaxial point groups 
, each of which is called a cyclic group, whose elements are 
. For a cyclic group where 
is even, one of its elements is equivalent to the symmetry operation .
Let’s suppose the object in figure I have complete arrow heads (see figure IV). Other than and , the object has a plane of symmetry perpendicular to the axis of rotation (the horizontal plane is denoted by 
) and a centre of inversion 
at the origin. The set of symmetry operations 
forms the Abelian point group 
(see multiplication table iii). The object in figure IVa belongs to the 
point group (see multiplication table iv) with the set of elements 
, where 
. Similar to the 
point groups, if we apply the same logic to other related objects, we have an infinite number of 
point groups, each with symmetry operations of 
if is even (one of the 
symmetry operations is equivalent to ) and 
if is odd.
Question
Why aren’t 
and 
elements of the group 
?
Answer
They are not unique elements, as they are equivalent to other ‘simpler’ elements of the group:
For an object that is made up of two equally spaced equilateral (or isosceles) triangles on a circle (see figure V above), we have the Abelian point group 
, whose elements are the symmetry operations , 
, 
and 
, where the symmetry elements for and 
are the vertical planes: 
-plane and 
-plane respectively. The multiplication table for this point group is shown in table v above. As before, we have an infinite number of 
point groups. The cone depicted in figure VI is an example of an object that belongs to the point group, where 
, i.e. the 
point group, with the set of symmetry operations: ,C_2,C_{\infty}^-(\phi),\sigma_v(\varphi)\})
.
The next few related point groups are 
and 
(for dihedral). They are related in the sense that they have in common the elements 
and 
, which serve as identifiers in categorising molecules by point groups.
The 
and 
point groups are identical to the 
and point groups respectively in non-standard orientation, i.e. the principal axis is along the 
-axis. A dumbbell has symmetry elements that are associated with the symmetry operations of
,C_{\infty}^-(\phi),C_{\infty}^{+'}(\phi),C_{\infty}^{-'}(\phi),\sigma(\varphi),i,S_{\infty}^+(\phi),S_{\infty}^-(\phi)\})
which is a special point group like 
.
Next, we have the 
point group, which is in general associated with the symmetry operations 
. When 
, we have the point group 
, which is the same as the point group 
. When 
, the point group 
is identical to the point group 
. For 
, we need to analyse the point group with odd and even separately.
Consider an point group 
. When is odd, the symmetry operation 
. If 
, then we have 
or 
(according to the closure property of a group). Since 
, then 
. Moreover, 
, which implies that 
. We can rewrite the symmetry operations of the point group ( is odd) as
which is equivalent to the set of elements of the point group . For example, the object in figure IVa belongs to the point group 
and hence to the point group .
When is even, the symmetry operation 
. Since 
, we have or . Similarly, 
or 
. This implies that 
. The elements of 
can be denoted by 
, where 
can be odd or even. If is even, then 
. For example, 
and 
. If is odd, then 
. We can therefore express the symmetry operations for the point group (when is even) as 
. Figure VII depicts an object that belongs to the 
point group.
Taking into account the above characteristics of the point group, it is possible to relabel it as the 
point group, where 
.
The rest of the point groups are the tetrahedral groups 
, the octahedral groups 
, the icosahedral groups 
and the special orthogonal group in 3-dimensions )
(also known as the full rotation group). The tetrahedral and octahedral groups are collectively called the cubic groups.
| Symbol |
Object |
Group elements |
Notes |
 |
 |
 |
Each of the three axes passes perpendicularly through the centre of one of the three depicted faces, e.g.  . Each of the four  axes passes through one of four body diagonals, e.g.  . |
 |
 |
or simply
|
Same rotation axes as . Three  axes (each with 2 symmetry operations:  ) coincident with the axes. Each of the six  passes through two diagonally opposite edges of the cube. |
 |
 |
 |
Same rotation axes as . Four  axes (each with 2 symmetry operations:  ) coincident with the axes. A centre of symmetry and three : i) bisecting  and  , ii) bisecting  and  , iii) bisecting  and  . |
 |
 |
or simply
|
Same as , but the axes are now  axes (each with 3 symmetry operations:  ). Six  axes through mid-points of diagonal edges, e.g.  and . |
 |
 |
 |
Same as , but includes centre of inversion , axes of  and mirror planes of and  . |
 |
 |
 |
The object is a snub dodecahedron with 92 faces (12 pentagons, 80 equilateral triangles), 150 edges and 60 vertices. |
 |
 |
 |
The object is a truncated icosahedron with 32 faces (12 pentagons, 20 hexagons), 90 edges and 60 vertices. Same symmetry elements as with the addition of a centre of inversion, improper axes and mirror planes.
|
|
 |
, all possible rotations |
The object is a sphere with an infinite number of rotation axes, each with all possible values of . |
