Infinitely many not locally soluble $SI^*$-groups

Abstract

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 also they 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 example of a not locally soluble $SI^*$-group built by Hall and by Kov\'acs and Neumann.