On simple Kn-groups for n = 5, 6

A finite nonabelian simple group is called a simple K n -group if the order of G has exactly n distinct prime factors. M. Herzog and W. J. Shi gave a characterization of simple K n -group for n = 3, 4, respectively. In this paper, we characterize all simple K n -groups for n = 5, 6. Introduction First we need some notation. Given a natural number n and a finite simple group G , we denote by π( n ) and Π( G ) the number of distinct prime factors of n and the set of distinct prime factors of | G |, respectively. We say that G is a simple K n -group if |Π( G )| = n . Also when a, b are two natural numbers, by ( a, b ) we mean gcd( a, b ). The rest of notation is standard and you can find them for example in. Huppert in studied the following conjecture: Conjecture 1 Let H be a finite nonabelian simple group and denote by cd(H) the set of the degrees of the irreducible complex characters of H. If cd(H) = cd(G) for some finite group G, then G ≅ H × A with A abelian . He proved this conjecture for some H by the following procedure: In the first step he showed that G ′ = G ″. In a second step he proved that whenever G ′/ M is a chief factor of G , then G ′/ M ≅ H .