Linear groups with the minimal condition on subgroups of infinite central dimension. (Q1879640)

Let \(V\) be a vector space over a field \(F\). A subgroup \(G\) of the general linear group \(\text{GL}(V)\) on \(V\) is said to have infinite central dimension if \(\dim_F(V/C_V(G))\) is infinite. The authors study subgroups \(G\) of \(\text{GL}(V)\) of infinite central dimension whose set of subgroups with infinite central dimension satisfies the descending chain condition. They have a number of nice results. For example, if \(G\) is also locally finite, then \(G\) is almost soluble. As a second example, if \(G\) is also almost locally soluble, then \(G\) is again almost soluble. Further, in both cases, \(G\) satisfies the minimal condition on normal subgroups, if \(\text{char\,}F=0\) then \(G\) is a Chernikov group and if \(\text{char\,}F\neq 0\), then \(G\) is quite close to being a Chernikov group.