Question 13

1. By Lagrange's theorem, if $L$ is a subgroup of a finite group $G$, then the order of $L$ must divide the order of $G$:

   $$|L| \mid |G|.$$

2. Given $|G| = 15$, the positive divisors of $15$ are:

   $$\{1, 3, 5, 15\}.$$

3. We are given the following conditions on subgroup $L$:

   • $L$ is a proper subgroup: $L \ne G \implies |L| < 15$.
   • The size of $L$ is at least 4: $|L| \ge 4$.

4. Among the divisors $\{1, 3, 5, 15\}$, the only divisor that satisfies $4 \le |L| < 15$ is $5$.

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