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Infinitely many not locally soluble $$SI^*$$-groups. (English) Zbl 1171.20311
Summary: The class of those (torsion-free) $$SI^*$$-groups which are not locally soluble, has the cardinality of the continuum. Moreover, these groups are not only pairwise non-isomorphic, but they also generate pairwise different varieties of groups. Thus, the set of varieties generated by not locally soluble $$SI^*$$-groups is of the same cardinality as the set of all varieties of groups. It is possible to localize a variety of groups which contains all groups and varieties constructed. The examples constructed here continue the well known examples of a not locally soluble $$SI^*$$-group built by Hall and by Kovács and Neumann.

MSC:
 20F19 Generalizations of solvable and nilpotent groups 20E10 Quasivarieties and varieties of groups 20E15 Chains and lattices of subgroups, subnormal subgroups 20E07 Subgroup theorems; subgroup growth
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