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 єZ< ∞?

Printable View

- April 27th 2010, 08:05 PMMissMouseyModern 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**< ∞? - April 27th 2010, 08:39 PMTinyboss
The order of a subgroup divides the order of the group. Groups of prime order are cyclic.

- April 27th 2010, 10:41 PMMeow
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.

- April 27th 2010, 10:59 PMaliceinwonderland
Well, I might be wrong, but the condition should be "proper" subgroup. There are groups of order pq that are non-abelian.

- April 28th 2010, 07:57 AMMissMouseyThank you everyone
I've solved the problem. (Nod)