Dashkova O. The structure of infinite dimensional linear groups and modules over group rings.

The thesis is devoted to study of infinite dimensional linear groups with restrictions on some systems of subgroups and modules over group rings of commutative Noetherian rings. It was described the structure of an infinite dimensional linear solvable group of infinite central (fundamental) dimension and of infinite rank such that each proper subgroup of infinite rank has a finite central (fundamental) dimension for a sectional p-rank, 0-rank, an abelian sectional rank, a special rank of a group. It is proved the solvability of an infinite dimensional linear locally solvable group of infinite central (fundamental) dimension and of infinite rank such that each proper subgroup of infinite rank has a finite central (fundamental) dimension for a sectional p-rank, 0-rank, an abelian sectional rank, a special rank of a group. The analogy of a central dimension of an infinite dimensional linear group for modules over group rings is introduced. It is proved the solvability of a locally solvable group G if A is a faithful ZG-module, Z is a ring of integers, a group G satisfies the minimality condition for subgroups such that their cocentralizers in the module A are not Artinian Z-modules. The analogous result is obtained for a ring of p-adic integers. It is proved that a group G is isomorphic to a quasicyclic group Cq for some prime number q if A is a faithful ZG-module, a group G is locally solvable, the cocentralizer of each proper subgroup of a group G in the module A is an Artinian Z-module and the cocentralizer of a group G in the module A is not an Artinian Z-module. The analogous result is obtained for a ring of p-adic integers.