Recognition of some symmetric groups by the set of the order of their elements. (Q1878568)

Denote by \(\pi_e(G)\) the set of orders of the elements of a finite group \(G\). One says that \(G\) is characterizable if \(\pi_e(G)=\pi_e(H)\) for a finite group \(H\) implies \(G\simeq H\). It has been conjectured that for all primes \(p\geq 7\) the symmetric groups \(\text{Sym}(p)\) and \(\text{Sym}(p+1)\) (excluding Sym(8)) are characterizable. The authors show that this conjecture is true for primes \(p\) between 50 and 100.