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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.
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