The Burnside problem and identities in groups
著者
書誌事項
The Burnside problem and identities in groups
（Ergebnisse der Mathematik und ihrer Grenzgebiete, 95）
Springer-Verlag, 1979
- : gw
- : us
- タイトル別名
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Проблема Бернсайда и тождества в группах
Problema Bernsaĭda i tozhdestva v gruppakh
Identities in groups
- 統一タイトル
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Problema Bernsaĭda i tozhdestva v gruppakh
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注記
Translation of: Проблема Бернсайда и тождества в группах, 1975
Bibliography: p. [301]-302
Includes indexes
内容説明・目次
内容説明
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
目次
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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