1.6: Constructing higher groups from simpler groups

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A group that contains a large number of symmetry elements may often be constructed from simpler groups. This is probably best illustrated using an example. Consider the point groups \(C_2\) and \(C_S\). \(C_2\) contains the elements \(E\) and \(C_2\), and has order 2, while \(C_S\) contains \(E\) and σ and also has order \(2\). We can use these two groups to construct the group \(C_{2v}\) by applying the symmetry operations of \(C_2\) and \(C_S\) in sequence.

\[\begin{array}{lllll} C_2 \: \text{operation} & E & E & C_2 & C_2 \\ C_S \: \text{operation} & E & \sigma(xz) & E & \sigma(xz) \\ \text{Result} & E & \sigma_v(xz) & C_2 & \sigma_v'(yz) \end{array} \tag{6.1}\]

Notice that \(C_{2v}\) has order \(4\), which is the product of the orders of the two lower-order groups. \(C_{2v}\) may be described as a direct product group of \(C_2\) and \(C_S\). The origin of this name should become obvious when we review the properties of matrices.