Finite Factor Groups of the Unimodular Group

Abstract

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: