Nonsolvable Groups with Three Nonlinear Irreducible Character Codegrees

For an irreducible characterχof a?nite group G,the codegree ofχis de-?ned as|G:ker(χ)|/χ(1).In this paper,the authors determine?nite nonsolvable groups with exactly three nonlinear irreducible character codegrees,which are L_(2)(2^(f))for f≥2,PGL_(2)(q)for odd q≥5 or M_(10).

Nonsolvable groups whose irreducible character degrees have special 2-parts

Let G be a nonsolvable group and Irr(G)the set of irreducible complex characters of G.We consider the nonsolvable groups whose character degrees have special 2-parts and prove that ifχ(1)_(2)=1 or∣G∣_(2)for everyχ∈Irr(G),then there exists a minimal normal subgroup N of G such that N≅PSL(2,2^(n))and G/N is an odd order group.

Nonsolvable D_2-groups

Let G be a finite group. Let Irrl（G） be the set of nonlinear irreducible characters of G and cdl（G） the set of degrees of the characters in Irr1（G）. A group G is said to be a D2-group if led1 （G）[ = ]Irr1 （G）I - 2. The main purpose of this paper is to classify nonsolvable D2-groups. Keywords Character degree, degree multiplicity, nonsolvable group

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.