| 非幂零真子群同阶类类数给定的有限群 |
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| 引用本文： | 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(中文版),2011,32(6):687-692. |
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| 作者姓名： | 史江涛 张翠 |
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| 作者单位： | 烟台大学数学与信息科学学院;Faculty of Mathematics;Natural Sciences and Information Technologies;University of Primorska;Glagoljaska 8;6000 Koper;Slovenia;Primorska Institute of Natural Sciences and Technology;Muzejski trg 2; |
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| 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 |
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| 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≌A_5或G≌SL_2(5).此外,完全分类了非平凡幂零子群同阶类类数≤5的非可解群和非平凡子群同阶类类数≤9的非可解群.
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| 关 键 词： | 非幂零子群 同阶类 可解群 |
Finite Groups in Which the Number of Classes of Non-nilpotent Proper Subgroups of the Same Order is Given |
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| SHI Jiangtao and ZHANG Cui.Finite Groups in Which the Number of Classes of Non-nilpotent Proper Subgroups of the Same Order is Given[J].Chinese Annals of Mathematics,2011,32(6):687-692. |
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| Authors: | SHI Jiangtao and ZHANG Cui |
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| Affiliation: | SHI Jiangtao~1 ZHANG Cui~2 1 School of Mathematics and Information Science,Yantai University,Yantai 264005,Shandong,China.] 2 Faculty of Mathematics,Natural Sciences and Information Technologies,University of Primorska,Glagoljaska 8,6000 Koper,Slovenia,Primorska Institute of Natural Sciences and Technology,Muzejski trg 2,Slovenia. |
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| Abstract: | 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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| Keywords: | Non-nilpotent subgroup Class of the same order Solvable group |
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