Infinitely many not locally soluble \(SI^*\)-groups.

*(English)*Zbl 1171.20311Summary: 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 KovĂˇcs 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 |