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Bulletin of the London Mathematical Society Advance Access originally published online on July 14, 2009
Bulletin of the London Mathematical Society 2009 41(5):804-816; doi:10.1112/blms/bdp056
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© 2009 London Mathematical Society

Automorphism invariance and identities

E. I. Khukhro

Sobolev Institute of Mathematics
Novosibirsk 630090
Russia

Ant. A. Klyachko

Faculty of Mechanics and Mathematics
Moscow State University
Moscow 119991
Russia
klyachko@mech.math.msu.su

N. Yu. Makarenko

Sobolev Institute of Mathematics
Novosibirsk 630090
Russia
and
Université de Haute Alsace
Mulhouse 68093
France
makarenk@math.nsc.ru

Yu. B. Melnikova

Faculty of Mechanics and Mathematics
Moscow State University
Moscow 119991
Russia
yuliamel@mail.ru

Received 9 December 2008. Revision received 26 March 2009.

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.

2000 Mathematics Subject Classification 17A36, 20E36 (primary), 17A30, 17A36, 20D15, 20E10, 20F50 (secondary).

The work of the second author was supported by the Russian Foundation for Basic Research (grant no. 08-01-00573). The work of the third author was supported by the Programme of Support of Leading Scientific Schools of the Russian Federation (grant NSh-344.2008.1).


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