Some Discussions On Connected Character Prime Divisor Graphs And Corresponding Finite Solvable Groups

For a given finite group G,the set of irreducible complex character degrees of G will be denoted by cd（G）.The character prime divisor graph Δ（G）has vertex set ρ（G）that consists of the primes that divide degrees in cd（G）;there is an edge between p and p if pq divides some degree a∈cd（G）.It is an important subject in group representation theory to study group structure by using character prime divisor graph.Domestic and foreign scholars have made a lot of important progress in the research on this subject.A finite group G is said to be connected if it’s character prime divisor graph Δ（G）is connected;otherwise,G is said to be a disconnected group.Accordingly,finite group G is said to be 1（2）-connected if it’s character prime divisor graph Δ（G）is 1（2）-connected.In particular,G is said to be a 1（2）-connected solvable group when G is solvable.In this paper,we study some connected character prime divisor graphs and corresponding finite solvable groups,and give the characterization of 1-connected solvable group with Fitting height 2.Furthermore,when the Sylow p-subgroup of G is normal for the cut vertex p of Δ（G）,we give the related structure information of G.Also,it is proved that these groups satisfy Gluck Conjecture,Isaacs-Navarro-Wolf Conjecture and Taketa’s inequality.The results are as follows:THEOREM 3.1 Let G be a finite solvable group and ρ（G）=π₁∪π₂∪{p} is a disjoint union,where πi≥1 for i=1,2.Assume that no prime in π₁ is adjacent in Δ（G）to any prime in π₂,and h（G）=2.Then the following conclusions hold:（1）Gis p-nilpotent and it’s normal p-complement is disconnected group with Fitting height 2.（2）There exists q∈π₁∪π₂ such that G has a normal abelian Sylow q-subgroup.If Assume q∈π₂,then π₂={q};（3）There exists s∈π₁ such that p is adjacent to s in Δ（G）;（4）Δ（G）is connected if and only if one of the following is true.（a）P as a Sylow p-subgroup of G is nonabelian;（b）There exists θ∈NL（Q）such that θis unstable under P,where Q is the unique Hall π₂-subgroup of G;（c）A×O_π1（G）（?）Z（G）,where A is the direct product of all normal abelian Sylow subgroups of G.THEOREM 3.2 Assume G is a 1-connected solvable group,p is the cut vertex ofΔ（G）and G has normal Sylow p-subgroup P.Let L be a Hall p’-subgroup of G.Then the following conclusions hold:（1）G/P’ is disconnected;（2）If G/P’ is not a disconnected group of the fourth type,then G=P×L,where L and G/P’ are of the same type;（3）If G/P’ is a disconnected group of the fourth class,let G/P’=V（?）H,where V and H correspond to "V" and "H" in the fourth type of the disconnected solvable groups.Then the following conclusions hold:①If V（?）P/P’,then G=P×L,and G is a disconnected group of the fourth type;②If V≤P/P’,then the following conclusions hold:（a）cd（G/Z）=cd（G）,where Z=C_L（P）≤Z（G）;（b）Let V=P₁/P’,G₁=P₁L,then either Δ（G₁）=Δ（G）has diameter 2 or Δ（G₁）is disconnected and G₁ is a disconnected group of the fourth or the sixth type.THEOREM 3.3 Let G be a 1-connected solvable group and suppose that G has normal Sylow p-subgroup P,where p is the cut vertex of Δ（G）.Then the following conclusions hold:（1）|G:F（G）|≤b（G）²;（2）V（G）≤F（G）;（3）dl（G）≤|cd（G）|.In addition,on the basis of the 1-connected character prime divisor graph,we further consider a class of 2-connected graph.We get theorem 4.1 and then prove that the Fitting height of the corresponding solvable group is at most 4.THEOREM 4.1 Let G be a solvable group and suppose that ρ（G）=π∪{p,q} is a disjoint union where |π|≥1.Assume that p is not adjacent in Δ（G）to q and G has an abelian Hall π-subgroup.Then G has Fitting height at most 4.THEOREM 4.2 Let G be a solvable group and ρ（G）=π₁∪π₂∪{p,q} is a disjoint union,where |π_i|≥1,i=1,2.Assume that p is not adjacent to q and no prime inπ₁ is adjacent in Δ（G）to any prime in π₂.Then G has Fitting height at most 4.