Question 54
Denote $\{1,2, \ldots, n\}$ by $[n]$, and let $S$ be the set of all permutations of $[n]$. Call a subset $T$ of $S$ good if every permutation $\sigma$ in $S$ may be written as $t_{1} t_{2}$ for elements $t_{1}$ and $t_{2}$ of $T$, where the product of two permutations is defined to be their composition. Call a subset $U$ of $S$ extremely good if every permutation $\sigma$ in $S$ may be written as $s^{-1} u s$ for elements $s$ of $S$ and $u$ of $U$. Let $\tau$ be the smallest value of $|T| /|S|$ for all good subsets $T$, and let $v$ be the smallest value of $|U| /|S|$ for all extremely good subsets $U$. Prove that $\sqrt{v} \geq \tau$.