Let G = S_n (the permutation group on n elements). Let A be a subset of G such that A generates G.

Is there an n-cycle g in G that can be expressed as

g = a_1 a_2 ... a_k

where a_i in A union A^{-1} and k is at most c_1 n^{c_2}, where c_1 and c_2 are constants?

What about 2-cycles, or elements of any other particular form?

Let $G = S_n$ (the permutation group on $n$ elements). Let $A\subset G$ such that $A$ generates $G$.

Is there an $n$-cycle $g$ in $G$ that can be expressed as

$g = a_1 a_2 ... a_k$

where $a_i\in A \cup A^{-1}$ and $k\leq c_1 n^{c_2}$, where $c_1$ and $c_2$ are constants?

What about $2$-cycles, or elements of any other particular form?