The restricted Burnside problem
In 1902, William Burnside wrote: "A still undecided point in the theory of discontinuous groups is whether the order of a group many not be finite while the order of every operation it contains is finite." Since then, the Burnside problem, in different guises, has inspired a considerable amount of research. One variant of the Burnside problem, the restricted Burnside problem, asks whether (for a given r and n) there is a bound on the orders of finite r-generator groups of exponent n. This book provides the first comprehensive account of the many recent results in this area. By making extensive use of Lie ring techniques it allows a uniform treatment of the field and includes Kostrikin's theorem for groups of prime exponent as well as detailed information on groups of small (3,4,5,6,7,8,9) exponent. The treatment is intended to be self-contained and as such will be an invaluable introduction for postgraduate students and research workers. Included are extensive details of the use of computer algebra to verify computations.
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The associated Lie ring of a group
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1)-Engel identity 3-group associated Lie rings associative algebra basic Lie products bracket products Burnside group characteristic combination of products commutator identities commutators of multiweight completes the proof coset enumeration define Engel identity entries exponent q expressed follows free group free Lie algebra groups of exponent hence homogeneous elements ideal identities which hold implies induction integer Jacobi identity Kostrikin's Theorem left-normed products Lemma Let G Lie element Lie identity linear combination linear span lower central series modulo multihomogeneous component multilinear identities nilpotent of class nilpotent quotient algorithm normal closure normal subgroup Note obtain p-group permutation power-commutator presentation product of commutators products of weight proof of Lemma proof of Theorem prove quotient group rank restricted Burnside problem rings of groups sandwiches satisfies the identity spanned subalgebra suppose trivial xr-j xu x2 zero