My class is studying on Sylow $p$-subgroups, and I had been stuck for several hours on determining the order of a Sylow $p$-subgroup of a group $G$ of finite order.
I asked a previous question like this here on this site but I completely misinterpreted my own question (by the way, this was my previous question: Determine order of a Sylow p-subgroup)
Now, I'd put up a bounty on that question as soon as the site allows me to (in $2$ days), but my quiz on this Sylow stuff is in $2$ days. Therefore, can I please try again with this question here? I'm going to try to make it easy for myself and make up a group order, say $|G| =12=2^2 \cdot 3$.
So let $G$ be a group of order $12$. For each prime $p$ dividing $|G|$, determine the order of a Sylow $p$-subgroup.
Now, I'd show my work, but I would not know the very first correct step (hence, my failed attempt at my previous question). All my "work" is shown there via the above link. Therefore any starting hints are nice.