有限交换环上极限线性群的Sylowp—子群  

Sylow p-subgroups of the Limit Linear Group GL Over Finite Commutative Rings

作　　者：刘绍武[1] 游宏[1] 

机构地区：[1]黑龙江大学,哈尔滨工业大学

出　　处：《数学进展》1996年第5期456-462,共7页Advances in Mathematics（China）

摘　　要：本文给出了有限交换局部环Ｒ上无限线性群ＧＬ（Ｒ）＝∪ｎＧＬｎＲ的Ｓｙｌｏｗｐ－子群的形式．令Ｍ是有限交换局部环Ｒ的唯一极大理想，ｋ＝Ｒ／Ｍ为Ｒ的剩余类域．用Ｘ（ｋ）表示ｋ的特征，并假定Ｐ与ｘ（ｋ）互素．作者证明了：ＧＬ（Ｒ）的任一Ｓｙｌｏｗｐ－子群Ｓ或者同构于的可数无限直积与Ｐ（ｊ）的无限直积的直积（当Ｐ≠２或Ｐ＝２，Ｘ（ｋ）β≡１（ｍｏｄ４））或者同构于Ｐｉ的无限直积与Ｐ（ｊ）的无限直积的直积（当Ｐ＝２，Ｘ（ｋ）β≡３（ｍｏｄ４）），这里，只是ＧＬ（ｅｐｉ）Ｒ（分别地，ＧＬ（２ｒｉ）Ｒ）的Ｓｙｌｏｗｐ－子群，Ｐ（ｊ））同构于Ｐ＝∪ｉ∈Ｉｐｉ，Ｉ是可数集．The present paper described the form of Sylow p-subgroups of the limit linear group GL(R)=∪nGLnR over a finite commutative local ring R. Let M be the unique maximal ideal of R,and k=R/M be the residue field of R. Denote by X(k) the characteristic of k and assume that p is prime to X(k). We proved that an. arbitrary Sylow p-subgroup S of the group GL(R)is either isomorphic to the direct product of the direct product of denumerable P and the direct product of infinite p(j) (when p≠2 or p = 2, X(k)β≡1(mod 4)) or is isomorphic to the direct product of the direct product of infinite Pi and the direct product of the infinite p(j) (when P = 2, X(k)β= 3(mod 4)), where Pi is the Sylow p-subgroup of GL(epi)R (Resp, GL(2ri)R), P(j)is isomorphic to P = ∪i∈IPi, I is a denumerable set.

关 键 词：无限线性群 局部环 SYLOW子群 交换环 线性群 

分 类 号：O152.3[理学—数学]