An alternative proof of a theorem on unimodular groups

Abstract

Introduction. Let G denote the multiplicative group of matrices /a b C where a, b, c, d are integers and ad bc= 1, and Go(n) the subgroup of G characterized by c_0 (mod n), where n is an integer different from 0. In a forthcoming paper [1] the author has proved the following theorem: THEOREM 1. Let H be a subgroup of G containing Go(n). Then H=Go(m), where m|n. The proof given was by an induction and made use of properties of the representatives of Go(nN) in Go(n). The referee for [1 ] furnished the author with an ingenious proof of Theorem 1 which avoided the induction. Since then the author has found a simpler proof which is more illuminating. This proof of Theorem 1 will be given here. Set