Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A4 / a / i

The molecular symmetry operations must be closed under composition. Write the proposed nonidentity operations as $R=C_2$ and $S=\sigma$. Their product $RS$ cannot equal any proposed member:Each conclusion contradicts the distinct operations stipulated. Therefore at least a fourth symmetry operation must exist. For a mirror plane containing the twofold axis, the product generates another mirror plane; for a plane perpendicular to it, the product is inversion. A tilted plane may generate still more operations, but cannot evade group closure.