Let T_(n) and S_(n) be the full transformation semigroup and the symmetric group on X_(n)={1,2,...,n},respectively.Let G be a transitiveimprimitive subgroupof S_(n) with nontrivial blocksΔand letαbe a transformation...Let T_(n) and S_(n) be the full transformation semigroup and the symmetric group on X_(n)={1,2,...,n},respectively.Let G be a transitiveimprimitive subgroupof S_(n) with nontrivial blocksΔand letαbe a transformation in T_(n)\S_(n).The kernel ofαis the partition of X_(n) induced by the equivalence relation{(x,y)|xα=yα};the kernel type ofαis the partition of n given by the sizes of the parts of the kernel.A transformation semigroup is called synchronizing if it contains a constant map.Then a group G synchronizes a transformationαif the semigroup(G,α)contains a constant map.In this paper,we study a transitive imprimitive permutation group G together with a non-invertible transformationαthat generate a synchronizing semigroup.We mainly discuss 7 cases where G synchronizes a special transformationαwith each kernel class A_(i)(A_(1)j)satisfying|A_(i)∩Δ|=1(|A_(1)j∩Δ|=1)for all blocksΔofG,that is,the kernel type ofαis(|A_(1)|,1,...,1),(|A_(1)1|,...,|A_(1m)|,|A_(2)|,...,|Ar|),or(|A_(1)|,...,|A_(t)|,1,...,1),or the rank is 2,3,4,or n-2.展开更多
In this paper, we investegate the intersection of a maximal intransitive subgroup with a maximal imprimitive subgroup. And, the structure of the second maximal intransitive subgroup of an alternating group is determined.
Based on a graph-theoretic analysis,we determine all the irreducible reflection subgroups of the imprimitive complex reflection groups G(m,p,n),and describe the irreducible subsystems of all possible types in the root...Based on a graph-theoretic analysis,we determine all the irreducible reflection subgroups of the imprimitive complex reflection groups G(m,p,n),and describe the irreducible subsystems of all possible types in the root system R(m,p,n)of G(m,p,n).展开更多
基金Supported by NSFC (No.12401024)the Scientific Research Innovation Project of Lingnan Normal University (Nos.LT2401,LT2410)。
文摘Let T_(n) and S_(n) be the full transformation semigroup and the symmetric group on X_(n)={1,2,...,n},respectively.Let G be a transitiveimprimitive subgroupof S_(n) with nontrivial blocksΔand letαbe a transformation in T_(n)\S_(n).The kernel ofαis the partition of X_(n) induced by the equivalence relation{(x,y)|xα=yα};the kernel type ofαis the partition of n given by the sizes of the parts of the kernel.A transformation semigroup is called synchronizing if it contains a constant map.Then a group G synchronizes a transformationαif the semigroup(G,α)contains a constant map.In this paper,we study a transitive imprimitive permutation group G together with a non-invertible transformationαthat generate a synchronizing semigroup.We mainly discuss 7 cases where G synchronizes a special transformationαwith each kernel class A_(i)(A_(1)j)satisfying|A_(i)∩Δ|=1(|A_(1)j∩Δ|=1)for all blocksΔofG,that is,the kernel type ofαis(|A_(1)|,1,...,1),(|A_(1)1|,...,|A_(1m)|,|A_(2)|,...,|Ar|),or(|A_(1)|,...,|A_(t)|,1,...,1),or the rank is 2,3,4,or n-2.
文摘In this paper, we investegate the intersection of a maximal intransitive subgroup with a maximal imprimitive subgroup. And, the structure of the second maximal intransitive subgroup of an alternating group is determined.
基金supported by National Natural Science Foundation of China(Grant Nos.10631010,10971138)the General Research Project of Shanghai Normal University(Grant No.SK200702)+2 种基金the Science Foundation of University Doctoral Project of China(Grant No.20060269011)Program for Changjiang Scholars and Innovative Research Team in University(Grant No.41192803)Shanghai Leading Academic Discipline Project(Grant No.B407)
文摘Based on a graph-theoretic analysis,we determine all the irreducible reflection subgroups of the imprimitive complex reflection groups G(m,p,n),and describe the irreducible subsystems of all possible types in the root system R(m,p,n)of G(m,p,n).
