GATE CS 2018 Q19 — Monoids and Groups

According to Lagrange's Theorem, the order (size) of any subgroup $H$ of a finite group $G$ must divide the order of $G$.

Given that the order of the group is $|G| = 84$:

• A proper subgroup $H$ of $G$ is a subgroup such that $H \neq G$, which implies $|H| < 84$.
• Therefore, the size of any proper subgroup $H$ must be a proper divisor of $84$.

The positive divisors of $84$ are:

The largest proper divisor of $84$ (strictly less than $84$) is $42$.

Thus, the size of the largest possible proper subgroup of $G$ is 42 (which is always realized, for example, when $G$ is the cyclic group $\mathbb{Z}_{84}$).