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
Let G be a finite group and H a subgroup of G.The normal index of H in G is defined as the order of K/H_(G),where K is a normal supplement of H in G such that|K|is minimal and H_(G)≤K■G.Let p be a prime which divide...Let G be a finite group and H a subgroup of G.The normal index of H in G is defined as the order of K/H_(G),where K is a normal supplement of H in G such that|K|is minimal and H_(G)≤K■G.Let p be a prime which divides the order of a group G.In this paper,some characterizations of G being p-solvable or p-supersolvable were obtained by analyzing the normal index of certain subgroups of G.These results can be viewed as local version of recent results in the literature.展开更多
This paper investigates the in uence of local SS-quasinormal maximal subgroups of Sylow subgroups on the structure of nite groups.We present several new criteria on p-nilpotency of nite groups by utilizing a small qua...This paper investigates the in uence of local SS-quasinormal maximal subgroups of Sylow subgroups on the structure of nite groups.We present several new criteria on p-nilpotency of nite groups by utilizing a small quantity of local SS-quasinormal maximal subgroups of Sylow p-subgroups.As applications,we obtain some sucient conditions for a nite group to be in a saturated formation containing the class of supersolvable groups.展开更多
Let G be a group and H;K be subgroups of G.H is called a TI-subgroup of G if H∩H^(g)=1 or H for every g∈G.K is called P-subnormal in G if there is a chain of subgroups K=K_(0)≤K_(1)≤K_(2)≤…≤K_(n-1)≤K_(n)=G suc...Let G be a group and H;K be subgroups of G.H is called a TI-subgroup of G if H∩H^(g)=1 or H for every g∈G.K is called P-subnormal in G if there is a chain of subgroups K=K_(0)≤K_(1)≤K_(2)≤…≤K_(n-1)≤K_(n)=G such that|K_(i):K_(i-1)|∈P for i∈{1;2;…;n}.Furthermore,K is called K-P-subnormal in G if there is a chain of subgroups K=K_(0)≤K_(1)≤K_(2)≤…≤K_(n-1)≤K_(n)=G such that either K_(i-1)is normal in Ki or|K_(i):K_(i-1)|∈P for i∈{1;2;…;n}.In this paper,some properties of a nite group in which some particular subgroups are TI-subgroups or P-subnormal subgroups or K-P-subnormal subgroups are given.展开更多
Lie algebras are special Leibniz algebras,so it is natural to view Lie algebras as Leibniz algebras.In this paper,we calculate all the Leibniz 2 cocycles of the Lie algebra K(1,0),which is helpful to classify one dime...Lie algebras are special Leibniz algebras,so it is natural to view Lie algebras as Leibniz algebras.In this paper,we calculate all the Leibniz 2 cocycles of the Lie algebra K(1,0),which is helpful to classify one dimensional central extensions of K(1,0)as Leibniz algebra.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.
基金Supported by the National Natural Science Foundation of China(Grant No.12071092)Guangdong Basic and Applied Basic Research Foundation(Grant No.2025A1515012072)+1 种基金the Natural Science Research Project of Anhui Educational Committee(Grant No.2024AH051298)the Scientific Research Foundation of Bozhou University(Grant No.BYKQ202419).
文摘Let G be a finite group and H a subgroup of G.The normal index of H in G is defined as the order of K/H_(G),where K is a normal supplement of H in G such that|K|is minimal and H_(G)≤K■G.Let p be a prime which divides the order of a group G.In this paper,some characterizations of G being p-solvable or p-supersolvable were obtained by analyzing the normal index of certain subgroups of G.These results can be viewed as local version of recent results in the literature.
基金Supported by NSF of China(12061011)NSF of Guangxi(2023GXN-SFAA026333)。
文摘This paper investigates the in uence of local SS-quasinormal maximal subgroups of Sylow subgroups on the structure of nite groups.We present several new criteria on p-nilpotency of nite groups by utilizing a small quantity of local SS-quasinormal maximal subgroups of Sylow p-subgroups.As applications,we obtain some sucient conditions for a nite group to be in a saturated formation containing the class of supersolvable groups.
文摘Let G be a group and H;K be subgroups of G.H is called a TI-subgroup of G if H∩H^(g)=1 or H for every g∈G.K is called P-subnormal in G if there is a chain of subgroups K=K_(0)≤K_(1)≤K_(2)≤…≤K_(n-1)≤K_(n)=G such that|K_(i):K_(i-1)|∈P for i∈{1;2;…;n}.Furthermore,K is called K-P-subnormal in G if there is a chain of subgroups K=K_(0)≤K_(1)≤K_(2)≤…≤K_(n-1)≤K_(n)=G such that either K_(i-1)is normal in Ki or|K_(i):K_(i-1)|∈P for i∈{1;2;…;n}.In this paper,some properties of a nite group in which some particular subgroups are TI-subgroups or P-subnormal subgroups or K-P-subnormal subgroups are given.
基金National Natural Science Foundation of China(11971315)。
文摘Lie algebras are special Leibniz algebras,so it is natural to view Lie algebras as Leibniz algebras.In this paper,we calculate all the Leibniz 2 cocycles of the Lie algebra K(1,0),which is helpful to classify one dimensional central extensions of K(1,0)as Leibniz algebra.
