| 一类不能作为自同构群的奇阶群 |
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| 引用本文： | 李世荣.一类不能作为自同构群的奇阶群[J].数学学报,1996,39(4):524-530. |
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| 作者姓名： | 李世荣 |
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| 作者单位： | 广西大学数学系 |
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| 基金项目： | 国家自然科学基金,广西自然科学基金 |
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| 摘 要： | 本文考虑如下问题：怎样的有限群可以作为另一个有限群的全自同构群？我们首先证明，若有限群Ｋ有一个正规Ｓｙｌｏｗｐ－子群使得｜Ｋ：Ｚ（Ｋ）｜ｐ＝ｐ２，那么Ｋ有２阶自同构．利用这个结果，我们证明了，若奇阶群Ｇ具有阶Ｐｓｍ（１≤ｓ≤３），ｐ为｜Ｇ｜的最小素因子，ｐｍ，ｍ无立方因子，则Ｇ不可能作为全自同构群．
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| 关 键 词： | 有限群，全自同构群，对合自同构 |
| 收稿时间： | 1994-9-19 |
| 修稿时间： | 1995-7-27 |
Some Groups of Odd Order Which Cannot Function as Automorphism Groups |
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| Li Shirong.Some Groups of Odd Order Which Cannot Function as Automorphism Groups[J].Acta Mathematica Sinica,1996,39(4):524-530. |
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| Authors: | Li Shirong |
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| Institution: | Li Shirong (Department of Mathematics, Guangxi University, Nanning 530004, China) |
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| Abstract: | The following problem is considered: what kind of finite groups can function as fullautomorphism group of a finite group? We first show that if the finite group K has a normalSylow p-subgroup such that |K/Z(K)|p=p2, then K has an automorphism of order 2. Usingthis result, we have shown that if G is an odd order group with order psm (1 ≤s ≤3), wherep is the smallest prime divisor of |G|, p m and m is cubefree, then G cannot function as fullautomorphism group. |
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| Keywords: | Finite group Full automorphism group Involution automorphism |
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