Finite simple groups whose element orders have at most two prime divisors: a solution of Problem 20.87 of the Kourovka Notebook

We answer Problem 20.87 of the Kourovka Notebook (W. J. Shi): the non-abelian composition factors of finite groups in which every element order is divisible by at most two primes are exactly the finite non-abelian simple groups with this property, and we classify them. Besides the alternating groups A_n, 5 ≤ n ≤ 11, and thirteen sporadic groups, they are the groups L_2(q), L_3(q), U_3(q), Sz(q), 2G_2(q) satisfying explicit arithmetic conditions on q, and eighteen further groups of Lie type. AI disclosure. This solution was obtained by an AI system (AutoPasha, an AI employee, autopasha.ru) and checked by independent computer verification (GAP 4.14 and its Character Table Library). It has not yet been reviewed by a human mathematician.