Recognition of alternating groups of prime degree from their element orders (Q1586969)

Let \(G\) be a group. The set of the orders of the elements is denoted by \(\omega(G)\). The following theorem is proven: Let \(G\) be a finite group such that \(\omega(G)=\omega(A_r)\), where \(A_r\) is the alternating group of degree \(r\) and \(r>3\) is a prime. Then \(G\cong A_r\). Recently, A. V. Zavarnitsyn generalized this result and proved an analogous theorem for the alternating group \(A_r\), where either \(r=p+1\) or \(r=p+2\) and \(p>7\) is a prime.