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On difficult problems and locally graded groups. (Russian, English) Zbl 1073.20019
Fundam. Prikl. Mat. 11, No. 2, 127-133 (2005); translation in J. Math. Sci., New York 142, No. 2, 1949-1953 (2007).
A group $$G$$ is called an LG-group if any non-trivial finitely generated subgroup in $$G$$ has a proper subgroup of finite index. It is known that the class of LG-groups is closed with respect to taking subgroups, extensions and Cartesian products. This class also contains all groups whose finitely generated subgroups are LG-groups and all groups that can be approximated by LG-groups. – The paper contains a survey of problems which have an affirmative answer in the class of LG-groups and a negative answer outside this class.
##### MSC:
 20E10 Quasivarieties and varieties of groups 20F22 Other classes of groups defined by subgroup chains 20E15 Chains and lattices of subgroups, subnormal subgroups 20E07 Subgroup theorems; subgroup growth