Automorphism invariance and identities

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 and useful facts. An example is produced showing that these results cannot be extended to arbitrary (non-multilinear) identities. As an application, a sharp estimate is given for the 'virtual derived length' of a (virtually solvable)-by-(virtually solvable) group. © 2009 London Mathematical Society