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

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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.