16 November 2008 - 18:37Result from Group Theory

In group theory Lagrange’s theorem is one of the most important in the theory of finite groups.

Theorem 2.6 (Lagrange’s theorem) The order of a subgroup H of a finite group G is a divisor of the order of G, i.e. |H| divides |G|.

One evident, but funny, implication of the theorem is in the answer of the following question: List all of the subgroups of any group whose order is a prime number.

Solution: According to Lagrange’s theorem, the order of a subgroup H of a group G must be a divisor of |G|. Since the divisors of a prime number are only the number itself and unity, the subgroups of a group of prime order must be either the unit element alone, H = {e}, or the group G itself, H = G, both of which are improper subgroups. Therefore, a group of prime order has no proper subgroups.