Large normal and characteristic subgroups satisfying outer commutator identities and their applications

We discuss recent results in which a normal subgroup of finite index (or with finite rank of the quotient, or some other well-behaving parameter) satisfying an outer (multilinear) commutator law is transformed into a large characteristic subgroup satisfying the same law. Here »large» means that the index (or the corresponding parameter) of the resulting characteristic subgroup is finite and bounded in terms of the index (or the corresponding parameter) of the original normal subgroup. Similar results also hold for ideals in arbitrary algebras over a field and, moreover, for a wider class of algebraic systems defined in terms of multioperator groups. We also give examples of applications of these results in various situations involving almost soluble groups, in particular, in the study of groups with almost fixed-point-free automorphisms. © 2009 World Scientific Publishing Co. Pte. Ltd.