On Finite Groups with Element Order Set of Width 3

Abstract

We study finite groups in which every element order is divisible by at most three distinct primes. We determine the possible nonabelian composition factors, prove that a nonabelian simple group with this property has at most $15$ prime divisors in its order, and deduce that $\rho (G)\le 192$ for every finite group $G$ with $\alpha (G)\le 3$.