How do you deduce the possible algebraic structures of a group from its order?

Question

For a finite group G with order 99, G contains the subgroups of order 1, 3 and 11 since they divide 99 and there exist a sylow p subgroup of order 9. Apart from this, what is the relation of order to direct products and isomorphism?
I dont know if I framed my question well but any help will be appreciated.

Answer 1

Let $G$ be a group of order $99$. From the Sylow theorems it follows that there is only one $3$-Sylow subgroup and only one $11$-Sylow subgroup. Let's say $P$ is a $3$-Sylow subgroup, $Q$ is an $11$-Sylow subgroup. Hence both $P$ and $Q$ are normal in $G$. More than that, $P\cap Q=\{e\}$ and since $|PQ|=\frac{|P||Q|}{|P\cap Q|}=\frac{9\times 11}{1}=99=|G|$ we conclude that $G=PQ$. So we have all the conditions of a direct product, so $G\cong P\times Q$.

Now, the only group of order $11$ up to isomorphism is $\mathbb{Z_{11}}$, and the only groups of order $9$ are $\mathbb{Z_9}$ and $\mathbb{Z_3}\times\mathbb{Z_3}$.Hence we get that $G\cong \mathbb{Z_9}\times\mathbb{Z_{11}}$ or $G=\mathbb{Z_3}\times\mathbb{Z_3}\times\mathbb{Z_{11}}$, so we actually found all groups of order $99$ up to isomorphism. Of course that can't be easily done for any order, but for order $99$ it is possible as you can see.