Let $G$ be a non-trivial group with no non-trivial proper subgroup. Prove that $G$ cannot be infinite group.

Let $G$ be a non-trivial group with no non-trivial proper subgroup. Prove that $G$ cannot be infinite group.

A hints is given as order of G is not infinite since $a$ and $a^{-1}$ are only generators. I know that infinite cyclic group has only two generators. But don't know how to prove the problem.

Take $1 \neq g \in G$. Then $\langle g \rangle \subseteq G$ is a subgroup. Since it is not the trivial subgroup and there are no proper subgroups it is equal to $G$. Thus $G$ is cyclic. Assume $G$ is infinite. Then the group $\langle g^2 \rangle$ is a proper subgroup.

Let $G$ be an infinite group, and $g \in G$ a non trivial element.

• What happens if $g$ has finite order?
• If $g$ has infinite order, then consider $\langle g^2\rangle$