According to Hall,a subgroup H of a group G is said to be pronormal if H and H^(g) are conjugate in<H,H^(g)>for every g∈G.In this survey,we discuss the role of pronormality for some subgroups of finite groups:H...According to Hall,a subgroup H of a group G is said to be pronormal if H and H^(g) are conjugate in<H,H^(g)>for every g∈G.In this survey,we discuss the role of pronormality for some subgroups of finite groups:Hall subgroups,subgroups of odd index,submaximal X-subgroup,etc.展开更多
A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to Hx in （H, Hx）. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is p...A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to Hx in （H, Hx）. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups （non-PRN-group all of whose proper subgroups are PRN-groups）. At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.展开更多
Let L be a simple Lie algebra with irreducible root system. having roots of different length, F be a field of charaCteristic different from 2, G = L(F) be a Chevalley group of type L over F. Denote by Φl the set of a...Let L be a simple Lie algebra with irreducible root system. having roots of different length, F be a field of charaCteristic different from 2, G = L(F) be a Chevalley group of type L over F. Denote by Φl the set of all long roots in Φl Set Gl = <xr (t); f∈EΦl,t∈F>. It is a subgroup of G generated by all the long root subgroups. This paper determines the pronormality of Gl in G when L is not of type G2.展开更多
基金supported by a NNSF Grant of China(Grant#11771409)Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciencesby SB RAS Fundamental Research Program I.1.1(project#0314-2016-0001).
文摘According to Hall,a subgroup H of a group G is said to be pronormal if H and H^(g) are conjugate in<H,H^(g)>for every g∈G.In this survey,we discuss the role of pronormality for some subgroups of finite groups:Hall subgroups,subgroups of odd index,submaximal X-subgroup,etc.
基金Supported by Natural Science Foundation of China （Grant No. 10871032）, Graduate Student Research and Innovation Program of Jiangsu Province （Grant No. CX10B-028Z）
文摘A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to Hx in （H, Hx）. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups （non-PRN-group all of whose proper subgroups are PRN-groups）. At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.
基金Project supported by the National Natural Science Foundation of China (No.19671079).
文摘Let L be a simple Lie algebra with irreducible root system. having roots of different length, F be a field of charaCteristic different from 2, G = L(F) be a Chevalley group of type L over F. Denote by Φl the set of all long roots in Φl Set Gl = <xr (t); f∈EΦl,t∈F>. It is a subgroup of G generated by all the long root subgroups. This paper determines the pronormality of Gl in G when L is not of type G2.
