Linear groups with rank restrictions on the subgroups of infinite central dimension.
The authors discuss four types of rank for a group \(G\). If \(G\) has a series whose factors are periodic or infinite cyclic, the 0-rank of \(G\) is the number of infinite cyclic factors in the series. If \(p\) is a prime the \(1_p\)-rank of \(G\) is the maximum of the ranks of the elementary Abelian \(p\)-sections of \(G\). Then \(G\) has finite 2-rank if every Abelian section of \(G\) has finite 0-rank and finite \(1_p\) rank for all primes \(p\). Finally, the 3-rank of \(G\) is the maximum of the minimal number of generators of \(H\) as \(H\) ranges over all the finitely generated subgroups of \(G\). (The notation here is not as in the paper under review.) Let \(V\) be a vector space over some field \(F\) and let \(G\) be a subgroup of \(\Aut_FV\). The authors define the central dimension of \(G\) to be the dimension (over \(F\)) of the space \(V/C_V(G)\). For rank meaning each of the four types of rank defined above the authors discuss locally soluble groups \(G\) of infinite rank and infinite central dimension such that every proper subgroup of \(G\) either has finite central dimension or has finite rank. For example, the authors claim that every such group is soluble and that in the case of 0-rank no such groups \(G\) exist.
scientific article; zbMATH DE number 5083160