I am trying to show that if $G$ is an abelian group of order 187, then the order of every nontrivial subgroup must be either order 11 or 17. I haven't learned the Sylow Theorems to be able to prove it using that method, but I have learned about Lagrange's Theorem, and I am not sure if it is applicable here.
Attempt: If $G$ is an abelian group of order 187, then by Lagrange's Theorem, every element in $G$ must have orders that are factors of 187, that is, 1, 11, 17, or 187. I am not exactly sure how to approach from here, but some help would be appreciated.
Best Answer
The order of a subgroup divides the order of a group. Since $187=11\cdot 17$, the only possible subgroups have order $1$, $11$, $17$, or $187$. (For an abelian group, all four possibilities will occur.)