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Locally (soluble-by-finite) groups with various restrictions on subgroups of infinite rank. (English) Zbl 1088.20013
The authors exhibit at first conditions for locally (soluble-by-finite) groups to be of finite rank. These are: (i) every Abelian subgroup has finite rank and the ranks of the torsion free Abelian subgroups are bounded by an integer $$r$$, (ii) every locally soluble subgroup has finite rank, (iii) the weak minimal condition is satisfied for locally soluble subgroups of infinite rank, (iv) same as (iii) but weak maximal condition.
On the other hand the authors show the following: If the group has infinite rank, it has infinitely many pairwise nonconjugate subgroups of infinite rank.

##### MSC:
 20E25 Local properties of groups 20E07 Subgroup theorems; subgroup growth 20E15 Chains and lattices of subgroups, subnormal subgroups 20F19 Generalizations of solvable and nilpotent groups
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