GATE CS 2014 Set 3 Q3 — Monoids and Groups

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

Given:

• The order of the group is $|G| = 15$.
• The divisors of $15$ are $1, 3, 5,$ and $15$. Therefore, the possible sizes for the subgroup $L$ are $1, 3, 5,$ or $15$.

We are also given the following constraints on $L$:

1. $L \neq G \implies |L| \neq 15$
2. The size of $L$ is at least $4 \implies |L| \ge 4$

The only divisor of $15$ that satisfies both conditions ($|L| \ge 4$ and $|L| < 15$) is $5$.

Thus, the size of $L$ is $5$.