## Abstract

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^{-1}x, and u_{n+1} = [xu_{n}x^{-1}, yu_{n}y^{-1}]. The main result states that a finite group G is solvable if and only if for some n the identity u_{n}(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 language | English |
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Pages (from-to) | 734-764 |

Number of pages | 31 |

Journal | Compositio Mathematica |

Volume | 142 |

Issue number | 3 |

DOIs | |

State | Published - 2006 |

## Keywords

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