# Math Help - A group with no subgroups

1. ## A group with no subgroups

Suppose that G is a group with more than one element and G has no proper, nontrivial subgroups. Prove that |G| is prime.

2. |G| must have no proper divisors, so...?

3. Then the |G| can only be divided by itself or 1. So it is a prime.

This is easy, too, should have gotten it earlier.

Thanks.

4. Originally Posted by tttcomrader
Suppose that G is a group with more than one element and G has no proper, nontrivial subgroups. Prove that |G| is prime.
Choose $x\in G$ so that $x\not = e$ which is possible by hypothesis. Construct the cyclic subgroup $\left< x\right>$. By hypothesis this must generate $G$ for this subgroup is nontrivial. Hence, $G$ is isomorphic to the cyclic group $\mathbb{Z}_n$. Now this group has proper non-trivial subgroups unless $n$ is a prime. Thus, $|G|=p$.