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Linear groups saturated by subgroups of finite central dimension. (English) Zbl 1454.20102

Summary: Let \(F\) be a field, \(A\) be a vector space over \(F\) and \(G\) be a subgroup of \(\mathrm{GL}(F,A)\). We say that \(G\) has a dense family of subgroups, having finite central dimension, if for every pair of subgroups \(H, K\) of \(G\) such that \(H\leqslant K\) and \(H\) is not maximal in \(K\) there exists a subgroup \(L\) of finite central dimension such that \(H\leqslant L\leqslant K\). In this paper we study some locally soluble linear groups with a dense family of subgroups, having finite central dimension.

MSC:

20H20 Other matrix groups over fields
20F22 Other classes of groups defined by subgroup chains
20E15 Chains and lattices of subgroups, subnormal subgroups
20F19 Generalizations of solvable and nilpotent groups
20E25 Local properties of groups
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