The Burnside Problem and Identities in Groups

Sergej I. Adian (Autor)

Buch | Softcover

XII, 314 Seiten

2011 | 1. Softcover reprint of the original 1st ed. 1979
Springer Berlin (Verlag)
978-3-642-66934-7 (ISBN)

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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

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.

Erscheint lt. Verlag	16.12.2011
Reihe/Serie	Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge
Übersetzer	John Lennox, James Wiegold
Zusatzinfo	XII, 314 p.
Verlagsort	Berlin
Sprache	englisch
Maße	170 x 244 mm
Gewicht	574 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Schlagworte	Burnsidesche Vermutung • Gruppe (Math.)
ISBN-10	3-642-66934-4 / 3642669344
ISBN-13	978-3-642-66934-7 / 9783642669347
Zustand	Neuware