JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (11): 11-15.doi: 10.6040/j.issn.1671-9352.0.2023.397

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Influence of traces of some subgroups on solvable groups

ZHANG Jia1,2, HE Jinlü1*, XIANG Yanhui1, WU Jinlian1   

  1. 1. School of Mathematics and Information, China West Normal University, Nanchong 637009, Sichuan, China;
    2. School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China
  • Published:2025-11-11

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Abstract: The influence of the supersolvable property of trace of primitive groups(which cannot be written as a proper intersection of subgroups)on the solvability of groups is analyzed. By dividing all maximal subgroups into two classes Fp (maximal subgroups whose indices are prime number), Fc (maximal subgroups whose indices are composite number), the connection between the nilpotency of the trace of every maximal subgroup in Fc and the structure of solvable groups is studied. By extending maximal subgroups to second maximal subgroups, three sufficient and necessary conditions on solvable groups are obtained.

Key words: maximal subgroup, primitive subgroup, second maximal subgroups, trace, solvable group

CLC Number: 

  • O152.1
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