For symmetric group $S_n$, we need to find a collection of subgroups $G_i$'s such that union of these subgroups is the group $S_n$ and each subgroup found is isomorphic to direct product of cyclic groups, each of order more than 1.
How can we find these subgroups such that sum of sizes of the subgroup is not much more than $S_n$ i.e. $(1 + o_n(1))\cdot|S_n|$?