= Solution
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:
$$
RS=E\Rightarrow R=S^{-1}=S,\qquad
RS=R\Rightarrow S=E,\qquad
RS=S\Rightarrow R=E.
$$
Each conclusion contradicts the distinct operations stipulated. Therefore \b[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.
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