A complete set of symmetry operations that can be performed on a molecule is called the Symmetry Group for that molecule, and satisfies key criteria for a mathematical group. This means that we can use the theorems of group theory (a branch of mathematics) to assist in problems of molecular symmetry. There are four key properties that a set of symmetry operations will obey if they belong to a group.
1. Probably the most important requirement for a set of symmetry operations to properly constitute a group is that if two operations in the group are multiplied together the product must also be an element (operation) in the group. In fact, every possible product of two operations in the set is also an operation in the group.
2. There is one operation, E, called the Identity Operation, that is the trivial one of making no change at all. If there is another operation, X, then:
E.X = X.E = X
3. Each of the operations has an inverse (reciprocal) which is a second operation which will exactly undo a given operation. So for operation R let us say that its inverse is S, then:
R.S = S.R = E
For example, for σ the inverse is σ, or σ x σ = σ2 = E
For a proper rotation, Cnm the inverse is Cnn-m or
Cnm x Cnn-m = Cnn = E
For i, i2 = E
4. The associative law holds, that is if X, Y, and Z are symmetry operations in the same space group then:
X.Y.Z = (X.Y) (Z) = X(Y.Z)
Note: [X.Y means "do symmetry operation Y first, followed by symmetry operations X"]
However, the commutative law, X.Y = Y.X does not always hold in groups. For water, the four operations do commute and such a group is said to be Abelian. All point groups that do not have an axis higher than two-fold are Abelian.
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