Abstract

The order components of a finite group were introduced in [5]. We prove that F4(q) is uniquely determined by its order components where q is an odd prime power. A main consequence of our result is the validity of Thompson's conjecture for the groups under consideration.

AMS Subject Classification: 20D05,20D60 Keywords : Finite group, simple group, prime graph, order component.

Introduction

If n is an integer, then π(n) is the set of prime divisors of n and if G is a finite group π(G) is defined to be π(|G|). The prime graph Γ(G) of a group G is a graph whose vertex set is π(G), and two distinct primes p and q are linked by an edge if and only if G contains an element of order pq. Let πi, i = 1, 2,…, t(G) be the connected components of Γ(G). For |G| even, π1 will be the connected component containing 2. Then |G| can be expressed as a product of some positive integers mi, i = 1, 2,…, t(G) with π(mi) = πi. The integers mi's are called the order components of G. The set of order components of G will be denoted by OC(G). If the order of G is even, we will assume that m1 is the even order component and m2,…, mt(G) will be the odd order components of G.