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Chapter 4: Problem 5

Write the multiplication table for \(\mathrm{C}_{2}\), point group.

Short Answer

Expert verified
The multiplication table for \(\mathrm{C}_{2}\) point group is: E ∗ E = E, E * \(C_2\) = \(C_2\), \(C_2\) * E = \(C_2\), and \(C_2\) * \(C_2\) = E.

Step by step solution

01

Identify the group elements

The first step in writing the multiplication table for \(\mathrm{C}_{2}\) point group is to identify the members of the group. The \(\mathrm{C}_{2}\) point group has just two members: the identity operation (E) and the two-fold rotation (\(C_2\)).
02

Create an empty table

Create an empty table with two columns and two rows. Label the rows and columns with the group elements: E, \(C_2\).
03

Fill in the diagonal

The intersection of the same group operations (or elements) is always the operation itself. So, the diagonal of the multiplication table which goes from the top left to the bottom right should be filled with the same operations which are E and \(C_2\). Now, the result of the operation E (no change) followed by any operation is just the second operation. Similarly, the result of any operation followed by E is just the first operation. So the first row (except for the identity) and the first column (except for the identity) of the multiplication table can now be filled in.
04

Complete the multiplication table

Finally, to complete the table fill the last position (the intersection of \(C_2\) and \(C_2\)). As the second \(C_2\) rotation is equivalent to no rotation (It will rotate twice and end at the starting point), the intersection of \(C_2\) and \(C_2\) (i.e., \(C_2 C_2\)) gives the identity operation E. Now, the multiplication table for \(\mathrm{C}_{2}\) is completed with the four entries E, \(C_2\), \(C_2\), and E.

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