Finite Groups in Which the Number of Classes of Non-nilpotent Proper Subgroups of the Same Order is Given
Finite Groups in Which the Number of Classes of Non-nilpotent Proper Subgroups of the Same Order is Given
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发表时间：
2011
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通讯作者：
Shi Jiangtao
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作者：
Shi Jiangtao
作为Schmidt定理的推广，得到如下结果：(1)具有少于3类同阶非幂零真子群的有限群是可解的；(2)G是具有恰好3类同阶非幂零真子群的不可解群当且仅当G≌A_5或G≌SL_2(5).此外，完全分类了至多含有5类同阶非平凡幂零子群的不可解群和至多含有9类同阶非平凡子群的不可解群.
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≌A_5 or G≌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.