GATE CS 2020 Q18 — 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 $|G| = 35$, the divisors of $35$ are:

Since we are looking for the largest possible size of a subgroup of $G$ other than $G$ itself (i.e., a proper subgroup), the size of the subgroup $|H|$ must be strictly less than $35$.

The possible sizes for a proper subgroup of $G$ are therefore $1, 5,$ and $7$.

The largest value among these possible sizes is $7$.

Note: Since $35 = 5 \times 7$ (where $5$ and $7$ are prime numbers), by Cauchy's Theorem, $G$ is guaranteed to have subgroups of order $5$ and $7$. Thus, a subgroup of size $7$ always exists.