Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L != G and that the size of L is at least 4. The size of L is __________.

GATE 2014 · Discrete Mathematics · Group Theory · medium

Answer: |L| = 5.

1. Apply Lagrange's theorem: Since L is a subgroup of G, |L| must divide |G| = 15. The divisors are 1, 3, 5, 15.
2. Filter by the constraints: L != G rules out 15. The condition |L| >= 4 rules out 1 and 3. Only 5 remains.