Around BurnsideSpringer Science & Business Media, 6 dec. 2012 - 222 sidor Perhaps it is not inappropriate for me to begin with the comment that this book has been an interesting challenge to the translator. It is most unusual, in a text of this type, in that the style is racy, with many literary allusions and witticisms: not the easiest to translate, but a source of inspiration to continue through material that could daunt by its combinatorial complexity. Moreover, there have been many changes to the text during the translating period, reflecting the ferment that the subject of the restricted Burnside problem is passing through at present. I concur with Professor Kostrikin's "Note in Proof', where he describes the book as fortunate. I would put it slightly differently: its appearance has surely been partly instrumental in inspiring much endeavour, including such things as the paper of A. I. Adian and A. A. Razborov producing the first published recursive upper bound for the order of the universal finite group B(d,p) of prime exponent (the English version contains a different treatment of this result, due to E. I. Zel'manov); M. R. Vaughan-Lee's new approach to the subject; and finally, the crowning achievement of Zel'manov in establishing RBP for all prime-power exponents, thereby (via the classification theorem for finite simple groups and Hall-Higman) settling it for all exponents. The book is encyclopaedic in its coverage of facts and problems on RBP, and will continue to have an important influence in the area. |
Innehåll
1 | |
4 | |
The Locally Nilpotent Radical | 10 |
Basic Conventions Elementary Combinatorics | 12 |
The Method of Sandwiches | 17 |
Filtrations in Lie Algebras | 23 |
Main results and Structure of Proofs | 26 |
Commentary | 28 |
Two Necessary Lemmas | 64 |
Sandwich Algebras | 67 |
Commentary | 79 |
Chapter 5 | 81 |
Chapter 4 | 83 |
Thick Pairs of Thin Sandwiches | 98 |
Completion of the Proof of the Main Theorem | 101 |
Commentary | 107 |
Chapter 2 | 31 |
Descent to Thin Sandwiches General Case | 33 |
Descent to Thin Sandwiches the Case p n | 39 |
Descent from C to Cp32 | 41 |
Commentary | 49 |
Chapter 3 | 50 |
A Second Footbridge Between Thin and Thick Sandwiches | 58 |
Evolution of the Method of Sandwiches | 108 |
Chapter 6 | 130 |
Chapter 7 | 164 |
Appendix I | 184 |
Appendix II | 197 |
214 | |
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a₁ arbitrary arguments assertion associative algebra assume B₁ Burnside group Burnside problem c₁ c₁₂ c₁c₂ c₁u² c₂ c₂c₁ cc₁ cc₂ Chap coc₁ commutators construction contains contradiction Corollary d₁ definition denote e₁ elements endomorphism Engel condition existence expression F of characteristic fact field F field of characteristic finite groups finite-dimensional follows formula free group free Lie algebra group of exponent i₁ induction j₁ k₁ L₁ Lemma Lie ring linear span locally nilpotent ideal Math monomials multilinear natural number nilpotency class nilpotent ideal notation obtained p-groups pair of thin permutation polynomial proof of Theorem Proposition 1.1 proved relations sandwich algebras sandwich Lie algebra sandwich of thickness satisfying the identity simple Lie algebra soluble subalgebra subsets summands Theorem 4.1 theory thick sandwich u₁ v₁ verbal ideal x₁ y₁ Zel❜manov