Combinatorial Group Theory: Presentations of Groups in Terms by Wilhelm Magnus

By Wilhelm Magnus

This seminal, much-cited account starts off with a pretty easy exposition of easy options and a dialogue of issue teams and subgroups. the subjects of Nielsen modifications, loose and amalgamated items, and commutator calculus obtain specific remedy. The concluding bankruptcy surveys be aware, conjugacy, and similar difficulties; adjunction and embedding difficulties; and extra. moment, revised 1976 version.

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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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