Navarro Vertices And Lifts Of Brauer Characters

It is a fundamental and important subject in the modular representation theory of finite groups to study the lifts of Brauer characters in p-solvable groups.In this thesis,we study three interrelated problems about lifts of Brauer characters.The first problem is concern with Cossey’s conjecture which uses vertices to give an upper bound on the number of lifts for a fixed irreducible Brauer character.The second problem focuses on the vertex version of the Fong-Swan theorem.Moreover,we examine the behavior of lifts of Brauer characters with respect to normal subgroups.At present,the Navarro vertex along with its deformation and generalization is the main technique used in these problems,and many achievements have been made for prime p>2,but there has been little progress for p=2.From a technical perspective,the principal obstacle is that the performance of vertex pairs is different from that of p>2.As a matter of fact,all the vertex pairs of a lift are neither linear nor conjugate when p=2.This thesis focuses on the case when p=2 while studying the three aforementioned issues.To overcome the technical difficulties encountered,we introduce the concept of twisted vertices.Besides,we prove that if a lift χ of an irreducible Brauer character has a linear Navarro vertex,all the vertex pairs for χ are linear,and simultaneously all of the twisted vertices for χ are linear and conjugate.This result,as our first major theorem,is a new technology and the starting point to use Navarro vertices to research the lifting problems of Brauer characters in the case where p=2.(Although a 2-solvable group is equivalent to a solvable group,we still formulate it as a p-solvable group with p=2 by convention in the literature.The default Brauer characters in this paper are about prime number p.)Theorem A Let G be a p-solvable group with p=2,and let χ∈ Irr(G)be a lift of some irreducible Brauer character.If χ has a linear Navarro vertex,then the following hold.(1)Every vertex pair for χ is linear.(2)All of the twisted vertices for χ are linear and conjugate in G.Using linear Navarro vertices,we construct a new reinforcement of the Fong-Swan theorem,which can be used to study the behavior of lifts of Brauer characters with respect to normal subgroups.Theorem B Let Q be a p-subgroup of a p-solvable group G,where p is a prime,and suppose that δ is a linear character of Q.If δ is stable in G,then restriction to p-regular elements defines a canonical bijection Irr(G|Q,δ)→ IBrp(G|Q),where Irr(G|Q,δ)is the set of all members in Irr(G)with Navarro vertex(Q,δ),and IBrp(G|Q)denotes the set of irreducible Brauer characters of G with vertex Q.As an application of Theorem A,we explore Cossey’s conjecture when p=2 and prove a weak form of the conjecture "one vertex at a time".In addition,we obtain a purely group-theoretic criterion for the conjecture.Theorem C Let G be a p-solvable group with p=2,and let φ∈ IBrp(G)with vertex Q.Then|Lφ(Q,δ)≤|NG(Q):NG(Q,δ)|for all δ∈Lin(Q),where Lφ(Q,δ)denotes the subset of lifts for φ with both a linear Navarro vertex and the twisted vertex(Q,δ),and NG(Q,δ)denotes the stabilizer ofδ in NG(Q).Based on Theorem A and Theorem B,we examine the performance of lifts of Brauer characters with respect to normal subgroups when p=2.Some sufficient conditions are given to ensure that the normal irreducible constituents of lifts remain lifts.Theorem D Let G be a p-solvable group with p=2,M?G,and suppose thatχ ∈ Irr(G)is a lift of some irreducible Brauer character with a linear Navarro vertex.Let(Q,δ)be a twisted vertex for χ,and write P=Q ∩ M and λ=δP.Assume one of the following conditions.(1)P is abelian and λ is invariant in NM(P).(2)P≤Op’p(M)and λ is invariant in NM(P).(3)λ(x)=λ(y)whenever x,y∈P are conjugate in M,that is to say,λ is stable in M.Then each irreducible constituent of χM is a lift.In order to use π-partial character theory,the new theory and technology created by Isaacs,we will study Navarro vertices and their applications of lifting problems inπ-separable groups.If π=p’,the complement of p in the set of all prime numbers,then π-partial characters of G are exactly Brauer characters(at p)of G.