# Homework Help: Show that a group with no proper nontrivial subgroups is cyc

1. Feb 20, 2017

### Mr Davis 97

1. The problem statement, all variables and given/known data
Show that a group with no proper nontrivial subgroups is cyclic.

2. Relevant equations

3. 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.

2. Feb 21, 2017

### 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 $\langle a\rangle=G$.