I was reading through a proof that no group of order $400$ is simple which can be found here: https://math.stackexchange.com/a/79644/169389
Here is an outline for a solution.
First of all, $|G| = 400 = 2^4 \cdot 5^2\ $. By Sylow's theorem we know that the number of Sylow 5-subgroups must be a divisor of $2^4$ and that it is $1$ modulo $5$. Thus it is either $1$ or $2^4$. If there is only one Sylow 5-subgroup, it must be normal.
For the other case, suppose first that the intersections of different Sylow 5-subgroups are always trivial. By counting elements you can conclude that $G$ has exactly one Sylow 2-subgroup, which is then normal.
If we have Sylow 5-subgroups $P$ and $Q$ such that $P \cap Q \neq \{1\}$, then $|P \cap Q| = 5$. Therefore $P \cap Q$ is normal in $P$ and $Q$, and thus is normal in the subgroup $\langle P, Q \rangle$ generated by $P$ and $Q$. Finally, show that either $\langle P, Q \rangle$ is normal in $G$ or equals $G$.
I am trying to get to grips with general problems like this and feel that this is one of the best explanations, but I still feel that this argument goes too quickly and I still need some clarification on a few things.
First problem: In the second paragraph we consider $n_5$(number of sylow 5-subgroups)$=2^4$ and suppose that the intersections of different Sylow $5$-subgroups $=1$.
When he says "by counting the elements", I guess he means the intersections of different Sylow $5$-subgroups which would be $1$ which we have assumed, how can we now conclude that $n_2=1$?
Second problem: I am wondering why $|P \cap Q| = 5$ and not $5$ or $25$, I think that it may be because they are different their intersection would have to be $5$ since if the intersection is only of order $25$ when the groups are the same and of order $25$, but if this could be clarified, it would be greatly appreciated
Third problem: To show $\langle P, Q \rangle$ is normal in $G$ or equals $G$. How did a commenter discern that the possible orders for $\langle P, Q \rangle$ are $50,100$ and $200$, why not $40$ or $25$?
Best Answer
(I) If $P_i, 1 \leq i \leq 2^4$ denote the distinct 5-Sylow subgroup of $G,$ then the number of non-identity element in $\cup_{1 \leq i \leq 2^4}P_i$ is $ 2^4 \cdot 24 = 384.$ The rest of the non-identity elements must belong to 2-Sylow subgroups and a 2-Sylow subgroup contains 15 non-identity elements. So it must be unique.
(II) $P \cap Q$ is a subgroup of $P$ and $Q.$ Since $|P| = |Q| = 25,$ the only choices for $|P \cap Q|$ is 1, 5, 25 (Lagrange's theorem). By assumption $|P \cap Q| \neq 1$ and $P, Q$ are distinct. So $|P \cap Q| = 5.$
(III) $\langle P, Q \rangle$ is a subgroup of $G$ and also contains both $P$ and $Q.$ So $|\langle P, Q \rangle|$ must divide $|G|$ and also must be divisible by $|P| = |Q|$ (Lagrange's theorem).