Groups with every proper quotient finite

Abstract

1·1. We shall write for the class of groups all of whose non-trivial normal subgroups have finite index. Thus, rather obviously, finite groups, simple groups, and the infinite cyclic and dihedral groups all lie in the class . Other examples of -groups are to be found in a variety of contexts. The main result of Mennicke (8) shows that, for n ≥ 3, the factor group of SL(n, Z) by its centre is a -group, where SL(n, Z) denotes the unimodular group of all n × n invertible matrices with integer coefficients and with determinant 1. In McLain(7), an example is given of an infinite, locally finite, locally soluble -group, whose only non-trivial normal subgroups are the terms of its derived series, and in (4), P. Hall discusses an infinite -group, all of whose proper quotients are finite p-groups. If is a class of groups closed under taking homomorphic images, it is easily seen that the existence of an infinite -group satisfying the maximal condition for normal subgroups implies the existence of an infinite -group in the class , so that certain questions concerning the finiteness of groups satisfying the maximal condition for normal subgroups can be interpreted as questions about -groups.