The Burnside problem and related topics.

*(English. Russian original)*Zbl 1230.20001
Russ. Math. Surv. 65, No. 5, 805-855 (2010); translation from Usp. Mat. Nauk 65, No. 5, 5-60 (2010).

This is a survey on the Burnside problem and the restricted Burnside problem (the latter called here the Burnside-Magnus problem), as well as other related problems in group theory. The historical and mathematical aspects of contributions of several outstanding mathematicians and their students are discussed in great detail. The timescale is from 1902, when the problem was stated by Burnside, to the present day, when new striking results continue to appear, but several problems still remain open.

Contents: 1. Historical sketch. 2 Short description of the Novikov-Adyan theory and first results. 3. Other properties of free periodic groups of odd exponent. 4. Infinite independent systems of group identities. 5. Non-commutative analogues of the group of rational numbers. 6. Periodic products of groups. 7. Some more results of other authors. 8. The Burnside-Magnus problem, effective estimates.

Contents: 1. Historical sketch. 2 Short description of the Novikov-Adyan theory and first results. 3. Other properties of free periodic groups of odd exponent. 4. Infinite independent systems of group identities. 5. Non-commutative analogues of the group of rational numbers. 6. Periodic products of groups. 7. Some more results of other authors. 8. The Burnside-Magnus problem, effective estimates.

Reviewer: Evgenii I. Khukhro (Nottingham)

##### MSC:

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

20F50 | Periodic groups; locally finite groups |

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

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |

20-03 | History of group theory |

01A60 | History of mathematics in the 20th century |