On Covering Number of Groups with Trivial Fitting Subgroup

Let G be a finite group and S be a subset of Irr(G).If for every prime divisor p of|G|there is a characterχin S such that p dividesχ(1),S is called a covering set of G.The covering number of G,denoted by cn(G),is defined as the minimal number of Card(S),where S is a covering set of G and Card(S)is the cardinality of set S.In this paper,we prove that if G is a finite group with F(G)=1,then the covering number cn(G)≤3.Especially,if PSL2(q)or J1 is not involved in G,then cn(G)≤2.