The Burnside problem and identities in groups
Author(s)
Bibliographic Information
The Burnside problem and identities in groups
（Ergebnisse der Mathematik und ihrer Grenzgebiete, 95）
Springer-Verlag, 1979
- : gw
- : us
- Other Title
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Проблема Бернсайда и тождества в группах
Problema Bernsaĭda i tozhdestva v gruppakh
Identities in groups
- Uniform Title
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Problema Bernsaĭda i tozhdestva v gruppakh
Available at / 64 libraries
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
ADI||3||1(L)||複本2645640
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Hokkaido University, Library, Graduate School of Science, Faculty of Science and School of Science図書
dc19:512.86/ad442021034406
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Note
Translation of: Проблема Бернсайда и тождества в группах, 1975
Bibliography: p. [301]-302
Includes indexes
Description and Table of Contents
Description
Three years have passed since the publication of the Russian edition of this book, during which time the method described has found new applications. In [26], the author has introduced the concept of the periodic product of two groups. For any two groups G and G without elements of order 2 and for any 1 2 odd n ~ 665, a group G @ Gmay be constructed which possesses several in- 1 2 teresting properties. In G @ G there are subgroups 6 and 6 isomorphic to 1 2 1 2 G and G respectively, such that 6 and 6 generate G @ G and intersect 1 2 1 2 1 2 in the identity. This operation "@" is commutative, associative and satisfies Mal'cev's postulate (see [27], p. 474), i.e., it has a certain hereditary property for subgroups. For any element x which is not conjugate to an element of either 6 1 or 6 , the relation xn = 1 holds in G @ G * From this it follows that when 2 1 2 G and G are periodic groups of exponent n, so is G @ G * In addition, if G 1 2 1 2 1 and G are free periodic groups of exponent n the group G @ G is also free 2 1 2 periodic with rank equal to the sum of the ranks of G and G * I believe that groups 1 2
Table of Contents
I. Basic Concepts and Notation.- 1. Words and Occurrences.- 2. Periodic Words.- 3. Aperiodic Words.- 4. Inductive Definitions.- 5. Symmetry and Effectiveness.- II. Periodic and Elementary Words of Rank ?.- 1. Periodic Words of Rank ?.- 2. Integral Words and Generating Occurrences of Rank ?.- 3. Minimal and Elementary Periods of Rank ?.- 4. Periodisation.- 5. Elementary Words of Rank ?.- 6. Local Properties of Normalisability.- 7. Completely Regular Occurrences of Rank ? - 1 and Further.- Properties of Elementary Words.- III. Reversals of Rank ?.- 1. Elementary Properties of Reversals.- 2. Stable Occurrences and Local Reversals of Rank ?.- 3. Cascades and Real Reversals.- IV. Reduced Words and Equivalence in Rank ?.- 1. Kernels and Reduced Words of Rank ?.- 2. Equivalance in Rank ?.- 3. Mutually Normalised Occurrences.- V. Coupling in Rank ?.- 1. Properties of the Operation of "Coupling".- 2. Further Properties of Equivalent Words.- VI. Periodic Groups of Odd Exponent.- 1. Existence of Infinite Periodic Groups of Odd Exponent.- 2. Systems of Defining Relations for a Free Group of Finite Exponent.- 3. Subgroups of a Free Group of Finite Exponent.- VII. Further Applications of the Method.- 1. Non-Abelian Groups in which Every Pair of Cyclic Subgroups Intersect Non-Trivially.- 2. Infinite Independent Systems of Group Identities.- References.- Index of Notation.
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