Finite Groups with Character Degrees of Two Distinct Primes

It is proved that if the degree of any non-linear irreducible character of a finite group G is a product of powers of two given distinct prime integers p and q, then G has an abelian Hall {p, q} subgroup H and an Abelian normal {p, q} complement A, and the centralizer in A of the Sylow p-subgroup of G is equal to the centralizer in A of the Sylow q-subgroup of G.

A Class of Finite Groups with Complete Character Degree Graphs

In this paper, we construct a new class of finite groups whose common divisor graphs are complete graphs, while there is no prime dividing all the nontrivial degrees.

Bounding Fitting Heights of Two Classes of Character Degree Graphs

In this article, we prove that a finite solvable group with character degree graph containing at least four vertices has Fitting height at most 4 if each derived subgraph of four vertices has total degree not more than 8. We also prove that if the vertex set ρ（G） of the character degree graph △（G） of a solvable group G is a disjoint union ρ（G） =π1∪π2, where ｜πi｜≥2 and pi,qi∈πi for i = 1,2, and no vertex in πl is adjacent in △（G） to any vertex in π2 except for p1p2 and q1q2, then the Fitting height of G is at most 4.

A Result on Character Degrees and Conjugacy Class Sizes in Finite Groups

Let G be a finite group and π be a set of primes including at least two elements. We write cd（G） and cs（G） to denote the set of complex irreducible character degrees and conjugacy class sizes of G , respectively, and write π（m）to denote the set of all prime divisors of a positive integer m . For any 1≠m∈cd（G） and 1≠m∈cs（G）, in this note, we shall present the corresponding group structures of finite group G in the case π（m）=π , respectively, which generalizes the result of finite groups with character degrees of two distinct primes. Furthermore, we shall see that the influence of the two sets on the group structure is analogous.

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.

Recognizing the Automorphism Groups of Mathieu Groups Through Their Orders and Large Degrees of Their Irreducible Characters＊

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders their character tables. and at most two irreducible character degrees of

A Graph Associated with |cd(G)| -1 Degrees of a Solvable Group

Let G be a group. We consider the set cd（G）／{m}, where m ∈ cd（G）. We define the graph △（G - m） whose vertex set is p（G - m）, the set of primes dividing degrees in cd（G）／{m}. There is an edge between p and q in p（G - m） ifpq divides a degree a ∈ cd（G）／{m}. We show that if G is solvable, then △（G - m） has at most two connected components.

A Characterization of Almost Simple K_3-groups

It is a well-known fact that characters of a finite group can give important information about the group＇s structure. Also it was proved by the third author of this article that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost simple group by using less information of its character table, and successfully characterize the almost simple K3-groups by their orders and at most three irreducible character degrees of their character tables.

On Huppert＇s Conjecture for Alternating Groups of Low Degrees

Bertram Huppert conjectured in the late 1990s that the nonabelian simple groups are determined up to an abelian direct factor by the set of their character degrees. Although the conjecture has been established for various simple groups of Lie type and simple sporadic groups, it is expected to be difficult for alternating groups. In [5], Hup- pert verified the conjecture for the simple alternating groups AN of degree up to 11. In this paper, we continue his work and verify the conjecture for the alternating groups of degrees 12 and 13.

Solvable D_2-Groups

Let G be a finite group, Irra（G） be the set of nonlinear irreducible characters of G and cdl （G） the set of degrees of the characters in Irrl （G）. A group G is said to be a D2-group if │cdl （G）│ = │Irrl（G）│ - 2. In this paper, we give a complete classification of solvable D2-groups.

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.