Combinatorial Optimization by Padberg M.W. (ed.)

By Padberg M.W. (ed.)

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Hence, since both x~:x~; ... x~~ and x~; ... x~~x~! are freely reduced, they are both cyclically reduced. -£) is a cyclic permutation of a(W z). CASE 4. , V =1= VI or € =1= -€1' but" = "r and € = €r. -£) is a cycI~c permutation of a(W z). This shows that a(TWzT-l) is a cyclic permutation of a(W z) when T has length one. Assume now as the inductive hypothesis that a(KW zK-l) is a cyclic permutation of a(W z). -E) is a cyclic permutation of a(W z). 3. SEC. 3 give a constructive solution to the word and transformation problem for a free group only when the free group is presented on free generators.

Thus, for example, to compute P(XIX2-IXaX3-1XzX2-1), we compute P(XIX2-IX3Xa-IX2X2-1) = X 1X 2- 1 • In general, p is defined inductively as follows: p(l) = 1, (€ = ±1; 11 = 1,2, ... ,n) and if p(U) = xZ~ ... xZ~ ('YJ;=±I; p;=1,2, ... 4 SEC. 35 ELEMENTARY PROPERTIES OF FREE GROUPS then oF v or if flq if flQ = rJq v and oF -E rJQ = -E· We first establish some properties of p from which the theorem follows easily. (a) p{ W) is freely reduced. (b) p(W) "':! W. (c) If V is freely reduced, then p(V) = V.

D) The group of n-by-n matrices with integer entries and determinant ± 1 (under multiplication). pz + q (e) The group of linear fractional transformations z ..... - - - where p, q, or, 8 are real numbers, p8 - qr =f. O. rz + 8 (f) The group (Xl' X 2 ' X 3 ' ••• ; X12, X22, X3 2 , ••• ). (g) The group (Xl' X 2 ' X 3 ' ••• ; xt', x{, Xa r , ••• ), r =f. O. (h) A finite non-cyclic group. 18. , the group az + b z ..... - - cz +d where a, b, c, d, are integers, ad - bc = 1. Moreover, let X = (z .....

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