| On Two Theorems of Finite Solvable Groups |
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| 作者姓名： | Shi Rong LI |
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| 作者单位： | Department of Mathematics, Guangxi University, Nanning 530004, P. R. China |
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| 基金项目： | Project supported by the National Natural Science Foundation of China （No.10161001） and the Natural Science Foundation of Guangxi of China （0249001） |
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| 摘 要： | For a finite group G, let T（G） denote a set of primes such that a prime p belongs to T（G） if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions： （1） G has a p-complement for each p∈T（G）; （2）│T（G）│= 2： （3） the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈T（G）; then G either is solvable or G/Sol（G）≌PSL（2, 7） where Sol（G） is the largest solvable normal subgroup of G.
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| 关 键 词： | 有限群 半群 群论 代数学 |
| 收稿时间： | 2003-07-01 |
| 修稿时间： | 2003-07-012003-11-14 |
On Two Theorems of Finite Solvable Groups |
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| Shi Rong LI.On Two Theorems of Finite Solvable Groups[J].Acta Mathematica Sinica,2005,21(4):797-802. |
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| Authors: | Shi Rong Li |
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| Institution: | (1) Department of Mathematics, Guangxi University, Nanning 530004, P. R. China |
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| Abstract: | For a finite group G, let τ(G) denote a set of primes such that a prime p belongs to τ(G) if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies
any one of the following conditions: (1) G has a p-complement for each p∈τ(G); (2) ∣τ(G)∣=2; (3)
the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈τ(G); then G
either is solvable or G/Sol(G)≅(2, 7) where Sol(G) is the largest solvable normal subgroup of (G).
Project supported by the National Natural Science Foundation of China (No. 10161001) and the Natural Science
Foundation of Guangxi of China (0249001) |
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| Keywords: | Finite solvable group The index of a maximal subgroup Simple group |
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