Abstract

A finite nonabelian simple group is called a simple Kn-group if the order of G has exactly n distinct prime factors. M. Herzog and W. J. Shi gave a characterization of simple Kn-group for n = 3, 4, respectively. In this paper, we characterize all simple Kn-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 Kn-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 1Let 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.