In this note,we show that if every maximal subgroup of a Sylow p-subgroup of a finite group has a p-solvable supplement then the group is necessarily p-solvable.This gives a positive answer to Problem 17.111 of the Ko...In this note,we show that if every maximal subgroup of a Sylow p-subgroup of a finite group has a p-solvable supplement then the group is necessarily p-solvable.This gives a positive answer to Problem 17.111 of the Kourovka Notebook(Unsolved Problems in Group Theory),which was posed by Skiba.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
基金supported by National Natural Science Foundation of China(Grant Nos.11101055 and 11171364)
文摘In this note,we show that if every maximal subgroup of a Sylow p-subgroup of a finite group has a p-solvable supplement then the group is necessarily p-solvable.This gives a positive answer to Problem 17.111 of the Kourovka Notebook(Unsolved Problems in Group Theory),which was posed by Skiba.
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.
