Download Word Problems II: The Oxford Books. Proceedings Oxford, 1976 by S.I. Adian, W.W. Boone and G. Higman (Eds.) PDF

By S.I. Adian, W.W. Boone and G. Higman (Eds.)

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Extra resources for Word Problems II: The Oxford Books. Proceedings Oxford, 1976

Example text

PjlR - 1 = ? 1. Suppose W = W , I - in G,. By virtue of Britton's lemma and the previous remark, we deduce that either 1 = 0 = r and RaKSo,or that 1, t > O and E , = S I , Ro = So%,,:lq;, . e. The proof of this assertion we need is completed by induction o n 1. em, We make some remarks that we shall use later on. Remark 1. reduced word W of the form (12) in which every R, is normal. , (for some Bxsx). Since E is ) . condition C 2 is equivalent to the normal and so E E C ( B ~ . , Remark 2. In Go, each word is transformed to its normal form by means of cancellations.

SB 2) @ = 0 S a < 7 and, for every a, all the relations in @a are of t h e form u@=, A p . ) We introduce the following notation: z, = u 20, P

Then Britton's lemma applied to (27) yields Bp,X21,= Y - in 21plh. It follows from Novikov's first theorem that X Y in K(2l). I). Inductively it suffices to Conversely suppose X consider the two cases: - XKA,X,, Y K X , B , and X F X I B , , Y K A , X , . Let, for example, the first of these occur. Then in 21pl, we have UplA,XipzX;A; U-' = piXiB,pzB TX: where U K(r:)-21xfi11:-1q;-1A :. The second case is dealt with similarly and so the theorem is proved. IPlh. The converse reduction was established in [7] and the following result proved.

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