Automorphism invariance and identities

TL;DR

This paper investigates how certain multilinear commutator identities that hold in large subgroups of groups or substructures of algebras also hold in large characteristic subgroups, with applications to estimating the virtual derived length of specific groups.

Contribution

It establishes that multilinear commutator identities extend from large subgroups to large characteristic subgroups, providing new insights into group and algebra structure theory.

Findings

Multilinear commutator identities hold in large characteristic subgroups.

Results do not extend to non-multilinear identities.

Provides a sharp estimate for the virtual derived length of certain groups.

Abstract

If an outer (multilinear) commutator identity holds in a large subgroup of a group, then it holds also in a large characteristic subgroup. Similar assertions are valid for algebras and their ideals or subspaces. Varying the meaning of the word "large", we obtain many interesting facts. These results cannot be extended to arbitrary (non-multilinear) identities. As an application, we give a sharp estimate for the `virtual derived length' of (virtually solvable)-by-(virtually solvable) groups.