一类不能作为自同构群的奇阶群  

Some Groups of Odd Order Which Cannot Function as Automorphism Groups

作　　者：李世荣[1] 

机构地区：[1]广西大学数学系

出　　处：《数学学报（中文版）》1996年第4期524-530,共7页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金;广西自然科学基金

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．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.

关 键 词：有限群 全自同构群 对合自同构 自同构群 奇阶群 

分 类 号：O152.1[理学—数学]