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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.

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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