10.32 (1986)

A group word is said to be universal on a group G if its values on $G$ run over the whole of $G$. For an arbitrary constant $c > 8/5$, J. L. Brenner, R. J. Evans, and D. M. Silberger (Proc. Amer. Math. Soc., 96, no. 1 (1986), 23–28) proved that there exists a number $N_0 = N_0(c)$ such that the word $x^r y^s$, $rs \neq 0$, is universal on the alternating group $\mathbb{A}_n$ for any $n \geqslant \max \{N_0, \ c \cdot \log m(r, s)\}$, where $m(r, s)$ is the product of all primes dividing $rs$ if $r, s \notin \{-1, 1\}$, and $m(r, s) = 1$ if $r, s \in \{-1, 1\}$. For example, one can take $N_0(5/2) = 5$ and $N_0(2) = 29$. Find an analogous bound for the degree of the symmetric group $S_n$ under hypothesis that at least one of the $r, s$ is odd: up to now, here only a more crude bound $n \geqslant \max \{6, \ 4m(r, s) - 4\}$ is known (M. Droste, Proc. Amer. Math. Soc., 96, no. 1 (1986), 18–22).