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Constituents And Fixed Points Of Automorphism Groups Of Block Designs

Posted on:2023-03-24Degree:DoctorType:Dissertation
Country:ChinaCandidate:J F ChenFull Text:PDF
GTID:1520306830983149Subject:Mathematics
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In this dissertation,we study automorphism groups of block designs.There are two main parts:(a)study flag-transitive automorphism groups with regular or Frobenius block constituents;(b)study some topics related to fixed points of automorphism groups and induced designs.The research of classifying flag-transitive 2-designs originates from the classification of flag-transitive 2-(v,k,1)designs(Buekenhout,etc.)in 1980s and 1990s.Since then,study on the classification of block designs satisfying other conditions became popular.For instance,in 2021,the classification of flag-transitive 2-designs satisfying(r,λ)=1 was almost finished.This is the extension of the classification of flag-transitive 2(v,k,1)designs.For other conditions,there are already many progresses.However,these classification results mainly focus on block designs with restricted parameters.There is still few result concerning restricting the permutation properties of automorphism groups.Classifying block designs with given block constituents is still a new research field.Except for 2-(v,k,1)designs,no systematic study are carried out yet.This research idea also originates from the study on subconstituents of primitive permutation groups.Wielandt,Wong,Quirin,Wang Jie,etc.finished the classification of primitive groups admitting a suborbit of length 2,3,4 or 5;Praeger and Wang jie also classified primitive groups with a 2-transitive subconstituent.These work shows us an idea that we may determine the integral structure of a primitive group by its local structure.Similarly,the block constituent of a automorphism group of a 2-design reflects the local properties of this automorphism group.Thus,classifying transitive block designs with given block constituent is a natural research idea.In the third chapter of this dissertation,we study flag-transitive symmetric designs with regular or Frobenius block constituents.We first prove that the Frobenius block constituent of a flag-transitive automorphism group of a symmetric design must be point subconstituent.In particular,we characterize automorphism groups with dihedral block constituents,and also prove that the block constituent must be point subconstituent.Moreover,prove that if D is a symmetric design with A prime,admitting a flag-transitive automorphism group G with cyclic block constituents,then either D is a(16,6,2)design or G is a Frobenius group.Afterwards,we study primitive groups whose point stabilizers are Frobenius groups or dihedral groups.These results are applied to reduce primitive automorphism groups of symmetric designs with Frobenius block constituents.At the mean time,study the condition under which the transitive automorphism groups with Frobenius block constituents must be primitive.Lastly,we give some examples.Obtain some families of 2-designs with regular or Frobenius block constituents by classifying flag-transitive 2-designs with Suzuki socle.The second work of this dissertation,i.e.,Chapter 4,studies some topics related to fixed points and induced designs of automorphism groups of symmetric designs or 2(v,k,1)designs.As applications,we give a characterization of finite projective geometry:a transitive symmetric design D(λ>1)having an axial automorphism of prime order((k-1)(λ-1)/λ)1/2 if and only if D≌PG2(3,p).Besides,we prove that if D is a 2-(v,k,1)design admitting an abelian automorphism group with two orbits on blocks,then D is either a projective plane,or a 2-(k2,k,1)design,or a point-transitive 2-(2k2-2k+1,k,1)design.
Keywords/Search Tags:2-design, automorphism group, flag-transitive, fixed point, constituent
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