Let \(G\) be a finite group on 84 elements. The size of a largest possible…

Key insight: subgroup orders must divide the group order (Lagrange's theorem).

• Compute divisors: 84 = 2^2 · 3 · 7, so any subgroup order must be a divisor of 84. The largest proper divisor of 84 is 42.

• Show 42 is attainable: take a group of order 12 that has a subgroup of order 6 (for example S3 × C2 or the dihedral group of order 12). The direct product C7 × (S3 × C2) has order 84 and contains C7 × S3 as a subgroup of order 7·6 = 42.

• Therefore the largest possible size of a proper subgroup of a group of order 84 is 42.