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

Question

Accepted Answer

The largest possible proper subgroup has 42 elements. By Lagrange's theorem, the order of any subgroup must divide the order of the group. A proper subgroup is strictly smaller than the whole group, so we want the largest divisor of 84 that is not 84 itself. The divisors of 84 in decreasing order are 84, 42, 28, ...; the largest proper one is 42.