How many operations are involved when n=3 in the symmetry operator Cn?

In the context of symmetry operations, the operator ( C_n ) represents a rotational symmetry around an axis, where ( n ) indicates the number of rotations that bring the molecule back into a congruent position within one full turn of 360 degrees. For ( n = 3 ), the ( C_3 ) symmetry operator involves rotating the object by ( 120^\circ ) increments, or ( \frac{360^\circ}{3} ).

When considering how many operations ( n = 3 ) entails, it’s important to recognize that there are three distinct rotational positions as a result of this symmetry. Each rotation by ( 120^\circ ) can be considered an individual operation, and after the third operation (which itself is a full rotation of ( 360^\circ), returning to the original position), the sequence is completed.

Thus, the total number of operations involved when implementing the ( C_3 ) symmetry operator is three: rotating the object first by ( 120^\circ ), then by ( 240^\circ ), and finally completing the 360° rotation to arrive back at the initial configuration. This highlights how the rotational symmetry encompasses these multiple configurations that the object can occupy