Soluble Groups

Abstract

The first version (historically) of the PSG Theorem to be proved dealt with the very special case of groups that are assumed to be both soluble and residually nilpotent; under these hypotheses, an elementary argument sufficed for the proof, involving none of the sophisticated mathematics that we have seen in Chapter 5. In Section 2 of this chapter we give a slightly more sophisticated (though still elementary) proof of the following sharper result: Theorem 9.1 Let G be a finitely generated virtually soluble group that is virtually residually nilpotent. Then either G has finite rank (and hence PSG), or there exist c > 1 and $d \in \mathbb{N}$ such that ${S_n}(G) \geqslant S_n^{ \triangleleft \triangleleft }(G) \geqslant {c^{{n^{1/d}}}}$ for all large n.