A non-cyclic one-relator group all of whose finite quotients are cyclic

Abstract

Let G be a group on two generators a and b subject to the single defining relation a = [ a, a b ]: . As usual [ x, y ] = x −1 y −1 xy and xy = y −1 xy if x and y are elements of a group. The object of this note is to show that every finite quotient of G is cyclic . This implies that every normal subgroup of G contains the derived group G ′. But by Magnus' theory of groups with a single defining relation G ′ ≠ 1 ([1], §4.4). So G is not residually finite. This underlines the fact that groups with a single defining relation need not be residually finite (cf. [2]).