A finitary Tits' alternative
For a group \(G\), let \(d(G)\) denote the minimal number of generators, and define the rank of~\(G\) to be \(\text{rk}(G)=\sup\{d(H)\mid H\) a subgroup of \(G\}\). In this note, the author proves a version of the well-known Tits alternative that applies to finite groups. Namely, given any family of finite groups of uniformly bounded rank, then either a subdirect product of these groups contains a non-cyclic free group, or each group contains a subgroup of bounded index whose derived subgroup is nilpotent of bounded class. This is based on a similar result dealing with a single finite group, where the first alternative is replaced by the existence of a finite approximation to the free group.
scientific article; zbMATH DE number 1295478