A footnote on residually finite groups (Q1916882)

The author gives a short and elementary proof of the following theorem. Let \({\mathfrak S}_n\) be the class of finite soluble groups of rank at most \(n\). Then a finitely generated residually-\(\mathfrak S_n\) group has a nilpotent normal subgroup modulo which the group can be viewed as a matrix group over a direct sum of fields. This result can then be used as a (weak) linearity criterion in place of the much deeper (and usually stronger) criterion of Lubotzky. In particular the theory of finitely generated groups with polynomial subgroup growth, the author claims, can now be developed without recourse to the theory of analytic pro-\(p\) groups.