My professor remarked in class today that the alternating group $A_5$ can be decomposed as a product of its Sylow subgroups but only if we align the Sylows in a specific way and combine them in the correct order. To be clear, by "product" of subgroups he meant it the sense that if $S$ and $T$ are subgroups of a group $G$, then their product is defined as $ST=\{st:s\in S{\text{ and }}t\in T\}$, and not in the sense of direct products or semi-direct products.
Now I'm not sure how to see this. I know that $A_5$ has:
5 2-Sylow subgroups of order 4 that are isomorphic to the Klein 4-group and which have a representative element as $\{(), (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(1, 3)\}$.
10 3-Sylow subgroups of order 3 that are isomorphic to the cyclic group $\mathbb Z_3$ and which have a representative element $\{(), (1, 2, 3), (1, 3, 2)\}$
6 5-Sylow subgroups of order 5 that are isomorphic to the cyclic group $\mathbb Z_5$ and which have a representative element $\langle (1, 2, 3, 4, 5)\rangle$.
(Edited) Can it be shown elegantly and explicitly that any element in $A_5$ may be written as product of some permutation in a $S_2 \in \mathrm{Syl_2}(A_2)$, some permutation in a $S_3 \in \mathrm{Syl_3}(A_2)$ and some permutation in a $S_5 \in \mathrm{Syl_5}(A_2)$ using the cycle notations? Even some hints might be helpful.