# Math Help - Prove

1. ## Prove

Suppose that G is a group that has exactly one nontrivial proper subgroup. Prove that G is cyclic and |G|=p^2 where p is prime.

2. Originally Posted by mandy123
Suppose that G is a group that has exactly one nontrivial proper subgroup. Prove that G is cyclic and |G|=p^2 where p is prime.
Let $H$ be the non-trivial proper subgroup.
Then $\{ e \} \subset H \subset G$.

Pick $a \in G - H$. And construct $\left< a \right>$.

3. ## Explain a little further

Could you please explain your proof a little more. Thanks.

4. Originally Posted by jonnyfive
Could you please explain your proof a little more. Thanks.
Note that once you construct $\left< a\right>$ it cannot be $H$ and cannot be $\{ e \}$. Since that is a subgroup and it has no other proper non-trivial subgroups it follows that $\left< a \right> = G$. Since the group is cyclic it is isomorphic to $\mathbb{Z}_n$ for some $n\geq 1$. By the properties of cyclic groups we can show that only $n=p^2$ are the ones with this unique non-trivial proper subgroup property.