Lie algebras and finite groups.

*(Russian)*Zbl 0547.17004This is a survey from a volume dedicated to the 50th anniversary of Mathematical Steklov Institute. It gives an account of the investigations on Lie algebras and finite groups in the works of the author, his colleagues from the Algebra Department of the Steklov Institute and their pupils. At the same time it provides an outline of the current state of the area (the works of other authors are also reviewed).

§1 ”Around Burnside” concentrates on the restricted Burnside problem which was solved in the affirmative by the author in 1959 for prime exponent. The brief history of this matter is also exposed. Firstly the connection between a finite group of prime exponent and its Lie ring is discussed. Secondly the author’s theorem on Engel Lie algebras and his ”sandwich method” is considered. The recent achievements on both themes inspired by the works of the author are included.

§2 is entitled ”Modular variations on the theme of Cartan”. It deals with the history and the current activities in the classification of simple Lie algebras over algebraically closed fields of characteristic \(p>0\). The program of this classification is described.

§3 ”The approaches to the main hypothesis. Representations and deformations of Lie algebras” is a continuation of the preceding one.

§4 is ”Automorphisms and cohomology of finite \(p\)-groups”. It is devoted to regular automorphisms, \(p\)-automorphic groups and homogeneous algebras, cohomology groups of nilpotent groups and algebras, Golod-Shafarevich theorem and the groups of Golod.

§5 is on ”Orthogonal decompositions of Lie algebras, lattices and automorphisms”. The objects from the title sometimes give nice realizations of finite simple groups, both of Lie type and sporadic.

The survey is written very densely and vividly and contains a good lot of names and references.

§1 ”Around Burnside” concentrates on the restricted Burnside problem which was solved in the affirmative by the author in 1959 for prime exponent. The brief history of this matter is also exposed. Firstly the connection between a finite group of prime exponent and its Lie ring is discussed. Secondly the author’s theorem on Engel Lie algebras and his ”sandwich method” is considered. The recent achievements on both themes inspired by the works of the author are included.

§2 is entitled ”Modular variations on the theme of Cartan”. It deals with the history and the current activities in the classification of simple Lie algebras over algebraically closed fields of characteristic \(p>0\). The program of this classification is described.

§3 ”The approaches to the main hypothesis. Representations and deformations of Lie algebras” is a continuation of the preceding one.

§4 is ”Automorphisms and cohomology of finite \(p\)-groups”. It is devoted to regular automorphisms, \(p\)-automorphic groups and homogeneous algebras, cohomology groups of nilpotent groups and algebras, Golod-Shafarevich theorem and the groups of Golod.

§5 is on ”Orthogonal decompositions of Lie algebras, lattices and automorphisms”. The objects from the title sometimes give nice realizations of finite simple groups, both of Lie type and sporadic.

The survey is written very densely and vividly and contains a good lot of names and references.

Reviewer: Evgenii I. Khukhro

##### MSC:

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

17Bxx | Lie algebras and Lie superalgebras |

20Dxx | Abstract finite groups |

17-03 | History of nonassociative rings and algebras |

01A65 | Development of contemporary mathematics |

20-03 | History of group theory |

01A74 | History of mathematics at institutions and academies (non-university) |

20F50 | Periodic groups; locally finite groups |

20F05 | Generators, relations, and presentations of groups |