有限交换环上线性群的Carter子群

Carter Subgroups of Linear Groups over Finite Commutative Rings

令Ｒ为有限交换局部环，Ｋ为其剩余类域．本文研究了Ｒ上一般线性群ＧＬｎＲ的Ｃａｒｔｅｒ子群的存在性及结构．得到的结果是：若ｃｈａｒＫ为奇数或Ｋ＝Ｆ２，ＧＬｎＲ中存在唯一的Ｃａｒｔｅｒ子群的共轭类，即Ｓｙｌｏｗ－２子群的正规化子；若ｃｈａｒＫ＝２且｜Ｋ｜＞２，ＧＬｎＲ中不含Ｃａｒｔｅｒ子群．

Let R be a finite commutative local ring and let M denote the unique maximal ideal of R and K denote the residue field of R. In the present paper, the author studied the existence and structure of the Carter subgroups of general linear groups GL nR over R. The main result is that if  char K is odd or K=F 2, there is a single conjugacy class of Carter subgroups in GL nR, namely, the normalizers of the Sylow 2-subgroups; if  char K=2 and |K|>2, GL nR contains no Carter subgroups.