On Beauville structures on the groups S n and A n

It is known that the symmetric group S _n , for n ≥ 5, and the alternating group A _n , for large n, admit a Beauville structure. In this paper we prove that A _n admits a Beauville (resp. strongly real Beauville) structure if and only if n ≥ 6 (resp n ≥ 7). We also show that S _n admits a strongly real Beauville structure for n ≥ 5.