Show that a group with no proper nontrivial subgroups is cyc

[Mr Davis 97]

Homework Statement

Show that a group with no proper nontrivial subgroups is cyclic.

Homework Equations

The Attempt at a Solution

If a group G has no proper nontrivial subgroups, then its only subgroups are $\{e \}$ and $G$. Assume that G has at least two elements, and let $a$ be any element besides $e$. Then $a$ generates a subgroup of $G$, but $G$ has no proper nontrivial subgroups, which means that $a$ must generate $G$, so $G$ is cyclic.

I feel that I am on the right track, but I also don't feel like I am being rigorous enough.

 

[andrewkirk]

Your proof is fine. I would just replace "$a$ generates a subgroup of $G$" by "$a$ generates a non-trivial subgroup of $G$", and observe that $\langle a\rangle$, the subgroup generated by $a$, cannot be proper, before stating that $\langle a\rangle=G$.