{"product_id":"9783642669347","title":"The Burnside Problem and Identities in Groups (Ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 2. Folge) (Reprint)","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 I. Basic Concepts and Notation.-\u003cbr\u003e1. Words and Occurrences.-\u003cbr\u003e2. Periodic Words.-\u003cbr\u003e3. Aperiodic Words.-\u003cbr\u003e4. Inductive Definitions.-\u003cbr\u003e5. Symmetry and Effectiveness.- II. Periodic and Elementary Words of Rank ?.-\u003cbr\u003e1. Periodic Words of Rank ?.-\u003cbr\u003e2. Integral Words and Generating Occurrences of Rank ?.-\u003cbr\u003e3. Minimal and Elementary Periods of Rank ?.-\u003cbr\u003e4. Periodisation.-\u003cbr\u003e5. Elementary Words of Rank ?.-\u003cbr\u003e6. Local Properties of Normalisability.-\u003cbr\u003e7. Completely Regular Occurrences of Rank ? - 1 and Further.- Properties of Elementary Words.- III. Reversals of Rank ?.-\u003cbr\u003e1. Elementary Properties of Reversals.-\u003cbr\u003e2. Stable Occurrences and Local Reversals of Rank ?.-\u003cbr\u003e3. Cascades and Real Reversals.- IV. Reduced Words and Equivalence in Rank ?.-\u003cbr\u003e1. Kernels and Reduced Words of Rank ?.-\u003cbr\u003e2. Equivalance in Rank ?.-\u003cbr\u003e3. Mutually Normalised Occurrences.- V. Coupling in Rank ?.-\u003cbr\u003e1. Properties of the Operation of \"Coupling\".-\u003cbr\u003e2. Further Properties of Equivalent Words.- VI. Periodic Groups of Odd Exponent.-\u003cbr\u003e1. Existence of Infinite Periodic Groups of Odd Exponent.-\u003cbr\u003e2. Systems of Defining Relations for a Free Group of Finite Exponent.-\u003cbr\u003e3. Subgroups of a Free Group of Finite Exponent.- VII. Further Applications of the Method.-\u003cbr\u003e1. Non-Abelian Groups in which Every Pair of Cyclic Subgroups Intersect Non-Trivially.-\u003cbr\u003e2. Infinite Independent Systems of Group Identities.- References.- Index of Notation.","brand":"Springer Verlag","offers":[{"title":"Default Title","offer_id":49173051244771,"sku":"00000_00000_00000_00000","price":54.99,"currency_code":"USD","in_stock":true}],"url":"https:\/\/usa.kinokuniya.com\/products\/9783642669347","provider":"Books Kinokuniya USA","version":"1.0","type":"link"}