| 史江涛,张翠.非幂零真子群同阶类类数给定的有限群[J].数学年刊A辑,2011,32(6):687~692 |
| 非幂零真子群同阶类类数给定的有限群 |
| Finite Groups in Which the Number of Classes of Non-nilpotent Proper Subgroups of the Same Order is Given |
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| DOI： |
| 中文关键词: 非幂零子群 同阶类 可解群 |
| 英文关键词:Non-nilpotent subgroup Class of the same order Solvable group |
| 基金项目:中国博士后科学基金(No20100470136,No201104027); “Agencija za raziskovalno dejavnost Republike Slovenije”,projmladi raziskovalci;“Agencija za raziskovalno dejavnost Republike Slovenije”,research program P1-0285资助的项目 |
| Author Name | Affiliation | E-mail | | SHI Jiangtao | School of Mathematics and Information Science,Yantai University,Yantai 264005,Shandong,China.Faculty of Mathematics,Natural Sciences and Information Technologies,University of Primorska,Glagoljaska 8,6000 Koper,Slovenia,Pr | shijt2005@163.com | | ZHANG Cui | Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska,
Glagolja\v{s}ka 8, 6000 Koper, Slovenia
Primorska Institute of Natural Sciences and Technology,
University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia. | cuizhang2008@gmail.com |
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| 中文摘要: |
| 作为Schmidt定理的推广, 证明了: (1) 非幂零真子群同阶类类数$<3$的有限群可解;
(2) $G$为非幂零真子群同阶类类数$=3$的非可解群当且仅当$G \cong A_5$或$G\cong \text{SL}_2(5)$.
此外, 完全分类了非平凡幂零子群同阶类类数$\leq 5$的非可解群和非平凡子群同阶类类数$\leq 9$的非可解群. |
| 英文摘要: |
| As an extension of Schmidt theorem, the following results are obtained: (1) A finite group with less than 3 classes of non-nilpotent
proper subgroups of the same order is solvable; (2) $G$ is a non-solvable group with exactly 3 classes of non-nilpotent proper
subgroups of the same order if and only if $G \cong A_5$ or $G\cong \text{SL}_2(5)$. Furthermore, non-solvable groups with at most 5
classes of non-trivial nilpotent subgroups of the same order and non-solvable groups with at most 9 classes of
non-trivial subgroups of the same order are completely classified. |
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