Identities for finite solvable groups and equations in finite simple groups

Tatiana Bandman, Gert Martin Greuel, Fritz Grunewald, Boris Kunyavskiǐ, Gerhard Pfister, Eugene Plotkin

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


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 u1 = x-2y-1x, and un+1 = [xunx-1, yuny-1]. The main result states that a finite group G is solvable if and only if for some n the identity un(x, y) ≡ 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).

Original languageEnglish
Pages (from-to)734-764
Number of pages31
JournalCompositio Mathematica
Issue number3
StatePublished - 2006


  • Finite solvable group
  • Gröbner basis
  • Identity
  • Lang-Weil estimate
  • Simple group
  • Trace formula


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