GATE 2018 CS – Question 29
Let $G$ be a finite group on 84 elements. The size of a largest possible proper subgroup of $G$ is ________.
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Correct answer: 42
Explanation
By Lagrange's theorem a proper subgroup has order dividing 84 and less than 84, so at most $\frac{84}{2}=42$. A group with a subgroup of order 42 exists, e.g. the cyclic group $\mathbb Z_{84}$.