This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post.
This problem is from assignment 5.
Let $H, K$ be subgroups of a group $G$ of orders 3,5 respectively. Prove that $H\cap K=\{1\}$.
The order of each element of $H$ must divide 3. Since the identity element is the only element with order 1, every other element in $H$ has order 3. Similar reasoning shows every nonidentity element of $K$ has order 5. Since $H$ and $K$ are both subgroups of $G$ they share the same identity element. Therefore, $H\cap K =\{1\}$.
Again, I welcome any critique of my reasoning and/or my style as well as alternative solutions to the problem.
Thanks.