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Identities for finite solvable groups and equations in finite simple groups

Published online by Cambridge University Press:  04 December 2007

Tatiana Bandman ,
Gert-Martin Greuel ,
Fritz Grunewald ,
Boris Kunyavskii ,
Gerhard Pfister  and
Eugene Plotkin
Tatiana Bandman
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, [email protected]
Gert-Martin Greuel
Affiliation:
Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, [email protected]
Fritz Grunewald
Affiliation:
Mathematisches Institut der Universität Heinrich Heine Düsseldorf, Universitätsstrße 1, 40225 Düsseldorf, [email protected]
Boris Kunyavskii
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, [email protected]
Gerhard Pfister
Affiliation:
Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, [email protected]
Eugene Plotkin
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, [email protected]
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Abstract

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We characterise the class of finite solvable groups by two-variable identities in a way similar to the characterisation of finite nilpotent groups by Engel identities. Let $u_1=x^{-2}y\min x$, and $u_{n+1} = [xu_nx\min,yu_ny\min]$. The main result states that a finite group G is solvable if and only if for some n the identity $u_n(x,y)\equiv 1$ holds in G. We also develop a new method to study equations in the Suzuki groups. We believe that, in addition to the main result, the method of proof is of independent interest: it involves surprisingly diverse and deep methods from algebraic and arithmetic geometry, topology, group theory, and computer algebra (SINGULAR and MAGMA).

Keywords

finite solvable groupidentitysimple groupGröbner basistrace formulaLang–Weil estimate20F1620F4520E1820D1013P1014G1514G1014-04
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 3 , May 2006 , pp. 734 - 764
Copyright
Foundation Compositio Mathematica 2006