GATE CS 2018 Q.29 (CS Q.19): Monoids and Groups

Let $G$ be a finite group on 84 elements. The size of a largest possible proper subgroup of $G$ is ________.

Your answer

Official answer and solution

Solution

Apply Lagrange's Theorem which states that the order of any subgroup must divide the order of the group.

Let $G$ be a finite group of order $|G| = 84$ . According to Lagrange's Theorem, if $H$ is a subgroup of $G$ , then $|H|$ must divide $|G|$ .

A proper subgroup $H$ of $G$ must satisfy $|H| < |G| = 84$ .

The divisors of 84 are $1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$ . The largest divisor strictly less than 84 is $\frac{84}{2} = 42$ .

Therefore, the maximum possible order of a proper subgroup of $G$ is 42.

Final answer

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