Let G be a finite group,and H a subgroup of G.H is called s-permutably embedded in G if each Sylow subgroup of H is a Sylow subgroup of some s-permutable subgroup of G.In this paper,we use s-permutably embedding prope...Let G be a finite group,and H a subgroup of G.H is called s-permutably embedded in G if each Sylow subgroup of H is a Sylow subgroup of some s-permutable subgroup of G.In this paper,we use s-permutably embedding property of subgroups to characterize the p-supersolvability of finite groups,and obtain some interesting results which improve some recent results.展开更多
We prove that a finite group G is p-supersolvable or p-nilpotent if some sub- groups of G are weakly s-semipermutable in G. Several earlier results are generalized.
A group G is said to have property μ whenever N is a non-locally nitpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property v satisfy property μ, where a grou...A group G is said to have property μ whenever N is a non-locally nitpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property v satisfy property μ, where a group G is said to have property v if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP（G） is the Hirsch-Plotkin radical of G, and φf （G） is the intersection of all the maximal subgroups of finite index in G （here φf（G） = G if no such maximal subgroups exist）. It is shown that a group G has property μ if and only if HP（G/φf（G）） = HP（G）/φf（G） and that the class of groups with property v is a proper subclass of that of groups with property it. Also, the structure of the normal subgroups of a group： nilpotency with the minimal condition, is investigated.展开更多
A subgroup A of a finite group G is called a local covering subgroup of G if A^(G)=AB for all maximal G-invariant subgroup B of A^(G)=(A^(G),g∈G).Let p be a prime and d be a positive integer.Assume that all subgroups...A subgroup A of a finite group G is called a local covering subgroup of G if A^(G)=AB for all maximal G-invariant subgroup B of A^(G)=(A^(G),g∈G).Let p be a prime and d be a positive integer.Assume that all subgroups of p^(d),and all cyclic subgroups of order 4 when p^(d)=2 and a Sylow2-subgroup of G is nonabelian,of G are local covering subgroups.Then G is p-supersolvable whenever p^(d)=p or p^(d)≤(√|G|_(p))or p^(d)≤|O_(p'p)(G)|_(p)/p.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 11201082 and 11171353)China Postdoctoral Science Foundation (Grant Nos. 2012M521724 and 2013T60866)Natural Science Foundation of Guangdong Province (Grant No. S201204007267)
文摘Let G be a finite group,and H a subgroup of G.H is called s-permutably embedded in G if each Sylow subgroup of H is a Sylow subgroup of some s-permutable subgroup of G.In this paper,we use s-permutably embedding property of subgroups to characterize the p-supersolvability of finite groups,and obtain some interesting results which improve some recent results.
基金supported by the National Natural Science Foundation of China(Nos.11771129,11301150,11601121)the Natural Science Foundation of Henan Province of China(No.162300410066)
文摘Let G be a finite p-group with a cyclic Frattini subgroup. In this paper, the automorphism group of G is determined.
基金Research of the authors is supported by NNSF of China （Grants 11171243 and 11001098）, Natural Science Foundation of Jiangsu （Grant BK20140451）, and University Natural Sci- ence Foundation of Jiangsu （Grant 14KJB110002）.
文摘We prove that a finite group G is p-supersolvable or p-nilpotent if some sub- groups of G are weakly s-semipermutable in G. Several earlier results are generalized.
基金Project supported by the National Natural Science Foundation of China （Nos. 11371335, 11471055）.
文摘A group G is said to have property μ whenever N is a non-locally nitpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property v satisfy property μ, where a group G is said to have property v if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP（G） is the Hirsch-Plotkin radical of G, and φf （G） is the intersection of all the maximal subgroups of finite index in G （here φf（G） = G if no such maximal subgroups exist）. It is shown that a group G has property μ if and only if HP（G/φf（G）） = HP（G）/φf（G） and that the class of groups with property v is a proper subclass of that of groups with property it. Also, the structure of the normal subgroups of a group： nilpotency with the minimal condition, is investigated.
基金Supported by the NSF of China(Grant No.11871011)。
文摘A subgroup A of a finite group G is called a local covering subgroup of G if A^(G)=AB for all maximal G-invariant subgroup B of A^(G)=(A^(G),g∈G).Let p be a prime and d be a positive integer.Assume that all subgroups of p^(d),and all cyclic subgroups of order 4 when p^(d)=2 and a Sylow2-subgroup of G is nonabelian,of G are local covering subgroups.Then G is p-supersolvable whenever p^(d)=p or p^(d)≤(√|G|_(p))or p^(d)≤|O_(p'p)(G)|_(p)/p.
