# Modern Algebra Problem

• Apr 27th 2010, 07:05 PM
MissMousey
Modern Algebra Problem
Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.

Can I say that pq = n є Z < ∞?
• Apr 27th 2010, 07:39 PM
Tinyboss
The order of a subgroup divides the order of the group. Groups of prime order are cyclic.
• Apr 27th 2010, 09:41 PM
Meow
For a slightly more descriptive answer, Lagrange's theorem says that for finite groups, a subgroup's order divides that of the group. Since the group's order is a product of two primes, a nontrivial subgroup should have order p, q or 1. Groups of prime order are cyclic, because in particular, the subgroup generated by any non-identity element must be the whole thing, as it would have order dividing a prime number.
• Apr 27th 2010, 09:59 PM
aliceinwonderland
Well, I might be wrong, but the condition should be "proper" subgroup. There are groups of order pq that are non-abelian.
• Apr 28th 2010, 06:57 AM
MissMousey
Thank you everyone
I've solved the problem. (Nod)