A group theoretic characterization of linear groups
The author gives a group-theoretical characterization of finitely generated linear groups over a field of characteristic zero. Let \(\Gamma\) be an abstract group. A \(p\)-congruence structure (with bound \(c\)) for \(\Gamma\) is a descending series of finite index normal subgroups of \(\Gamma =N_ 0\supseteq N_ 1\supseteq N_ 2\supseteq ...\) such that (i) \(\cap_{i=0}^{\infty} N_ i = \{1\}\); (ii) \(N_ 1/N_ i\) is a \(p\)-group for every \(i>1\); (iii) \(N_ i/N_ j\) can be generated by \(c\) elements for every \(j > i > 1\). Theorem. Let \(\Gamma\) be a finitely generated group. Then the following conditions are equivalent: (A) \(\Gamma\) is isomorphic to a matrix group over \(\mathbb{C}\). (B) \(\Gamma\) has a \(p\)-congruence structure for some prime \(p\).
scientific article; zbMATH DE number 4055902