Finite Factor Groups of the Unimodular Group
Finite Factor Groups of the Unimodular Group
复制标题
酉模群的有限因子群
DOI：
10.2307/1970380
复制
发表时间：
1965
影响因子：
4.9
通讯作者：
J. Mennicke
中科院分区：
文献类型：
--
作者：
J. Mennicke
1.设SL（n，Z）是所有有理整数系数的n × n-矩阵的群，det =+1。本文研究SL（n，Z）的有限因子群.当n = 2时，每个由两个二阶和三阶元生成的有限群都是SL（2，Z）的因子群。长期以来，人们一直认为，当n > 2时，情况完全不同。在下文中，除非另有说明，否则我们将假设n > 2。在该杂志最近的一篇论文中，J. L.布伦纳建立了以下的阶梯关系。设N是SL（n，Z）中的任意正规子群，不平凡，即，不是恒等子群或中心。取N的任意矩阵，设ja为g.c.d.。的非对角系数和对角系数的差异。让我来做g.c. d。N的所有元素的数Pa。记SL（n，Z）中包含I + me 21的最小正规子群Qn，，7（记法同[1]），设Nm*，.是与模m的纯量矩阵全等的所有矩阵的正规子群。那么下面的梯形关系成立：
1. Let SL(n, Z) be the group of all n x n-matrices with rationali nteger coefficients and det = + 1. In the present paper, we shall study the finite factor groups of SL(n, Z). For n = 2, every finite group which can be generated by two elements of order two and three is a factor group of SL(2, Z). It has long been conjectured that the situation is quite different for n > 2. In the following, we shall assume n > 2 unless otherwise stated. In a recent paper [1] in this journal, J. L. Brenner has established the following ladder relation. Let N be any normal subgroup in SL(n, Z), not being trivial, i.e., not being the identity subgroup or the center. Take any matrix of N, and let ja be the g.c.d. of the non-diagonal coefficients and the differences of diagonal coefficients. Let m be the g.c.d. of the numbers Pa for all elements of N. Denote by Qn,,7 the least normal subgroup of SL(n, Z) which contains I + me21 (the notation being as in [1]), and let Nm*,. be the normal subgroup of all matrices which are congruent to a scalar matrix modulo m. Then the following ladder relation holds: