Read e-book online Algebra VII: Combinatorial Group Theory Applications to PDF

By A. N. Parshin (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

ISBN-10: 3540637044

ISBN-13: 9783540637042

ISBN-10: 3642580130

ISBN-13: 9783642580130

From the experiences of the 1st printing of this ebook, released as quantity fifty eight of the Encyclopaedia of Mathematical Sciences:
"... This e-book may be very helpful as a reference and consultant to researchers and graduate scholars in algebra and and topology." Acta Scientiarum Mathematicarum, Ungarn, 1994 "... The e-book less than evaluation involves monographs on geometric features of staff idea: Combinatorial team thought and basic teams" by way of D.J.Collins and H.Zieschang ...: "Some difficulties of workforce concept concerning geometry" by way of R.I.Grigorchuk and P.F.Kurchanov. ... jointly, those articles shape a wide-ranging survey of combinatorial team conception, with emphasis a great deal at the geometric roots of the topic. this can be an invaluable reference paintings for the specialist, in addition to supplying an outline of the topic for the outsider or amateur. many various themes are defined and explored, with the most effects provided yet now not proved. this permits the reader to get the flavor of those issues with out turning into slowed down intimately. either articles supply accomplished bibliographies, in order that it truly is attainable to take advantage of this ebook because the start line for a extra distinct research of a selected subject of curiosity. ... In precis, a really fascinating ebook! Bulletin of the London Mathematical Society, 1996 "... In either essays the authors provide transparent and entire definitions, examples and statements (but now not proofs) of theorems, in order that the e-book might be understood by means of a reader with a minimum history in workforce concept or geometry. one of these reader, desiring to determine what's recognized during this zone, will locate this an entire and obtainable shop of information." modern Physics, 1994 "...This survey (Part II) offers for the 1st time that difficulties in monograph shape and incidentally deals a unifying remedy of many of the methods to their strategies, so far as they're recognized, including tricks to open difficulties. A titbit for each reader!" Monatshefte für Mathematik, 1995

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9). The disadvantage of the Reidemeister-Schreier method is that one has to know a system of coset representatives and must be able to determine the coset representative of an element. This is not possible in general. An alternative construction is the Todd-Coxeter method [Todd-Coxeter 1936]; in fact, essentially this method was used by K. 4. 11. Classification of n-Sheeted Coverings. Let G be a group. A homomorphism f : G --+ Sym( n) is called tmnsitive if to i, j E {I, ... , n} there is agE G such that f(g)(i) = j.

A parallel result, with the same caveat, holds for HNN-extensions. The normal form theorems yield a complete description of finite subgroups of amalgamated free products and HNN-extensions. 12. Theorem. LetG = (G I *G 2 : Al = A 2) orG = (Go,t I rlAIt = A2 ). Then any finite subgroup of G is conjugate to a subgroup of a factor or of the base group, as is appropriate. 0 In view of the parallels with free groups indicated by the normal form theorems, it is not surprising that any significant commutativity within an amalgamated free product or an HNN-extension is more or less confined to the factors or the base group as appropriate.

Proposition (a) A reduced product of r elements of a Nielsen reduced system (WI, ... , W n ) of elements has length at least r in the free group. Indeed 1+ IW· I) + r IW"I W"2 ... W"rl > ~(IW 2 IW"IW"2 ... W"rl > IW I 1 <: k <::: r. 1 't2 Zr - 'Z1 22 'lr - 21 'Zk' 'iT' 2 and ~- (b) A Nielsen reduced system (WI' ... ' W n ) of elements of the free group F = ( X I - ) freely generates a free subgroup of F. Proof. Consider a reduced product VI V2 ... Vn with Vk == Wi:k, and 10k = ±1. Now the Nielsen conditions (N1)-(N3) guarantee that, in cancelling any triple V k- I Vk Vk+I to reduced form in F, a central letter of Vk survives the cancellation process.

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Algebra VII: Combinatorial Group Theory Applications to Geometry by A. N. Parshin (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

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