Can a group always be written as a Zappa-Szep product of its Sylow subgroups? In particular, can a simple group (so that we don't have any proper normal subgroups) have an exact factorization, i.e. do there always exist $H, K<G$ with $H\cap K = \{1_G\}$ and $|H|\cdot|K|=|G|$?

To answer the question in the title, the group ${\rm PSU}(3,3)$ of order $6048 = 2^5.3^3.7$ cannot be wriiten as a product $P_1P_2P_3$ of $3$ Sylow subgroups in any order. That is proved in a paper I wrote with Petey Rowley a while ago:
D. Holt and P. Rowley, On products of Sylow subgroups in finite groups,
 Arch. Math. 60 (1993), 105-107.
The same group is a counterexample to the question in the body because you can check by computing all subgroups of ${\rm PSU}(3,3)$ on a computer that there are do not exist inters $a,b>1$ with $ab=6048$ such that this group has subgroups of orders $a$ and $b$. There may be smaller counterexamples.
