By Joan M. Holland (auth.)
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There is a wellestablished proof by induction that the property can be extended to four or more elements. Given a product of n elements it is permissible to partition it by brackets wherever we please provided we do not tamper with the sequence of the elements. Thus ab 2cde may be interpreted as {(ab)(bc)} (de) or as {a(b2)} {c(de)}. We shall need this property in later chapters. 4 find qr, ru, uv and verify that (qr)(uv)-=q{(ru)v}= {q(ru) }v. Also verify that { (ur) 2 }(v3)={ u(ru) }{ (rv)v 2 }.
The passive objects of the transformations no longer have the same algebraic form as the transformations. It is of interest that all six of the trigonometric ratios emerge after the change of symbol and that they can be permuted among themselves by the six transformations. The effect of t 5 can be concisely expressed as: cos 2 0~sec 2 0,-tan2 0~-cot2 0, cosec2 0+---+sin2 0. 8 Here are twelve expressions which form a group for substitution. They are not arranged in an orderly sequence. (2t-1)/(t+l), (1-2t)/(2-t), t, (t+l)/(2-t), 1/(1-t), (2-t)/(t+l), tf(t-1), 1ft, (t+l)/(2t-1), 1-t, 1-lft, (2-t)/(1-2t).
Is this a commutative group? 9 E\1)8 A C D E6s;6E E6;:6A Reference pentagon DC DC CD C B Fig. 8 The figures above show p as a clockwise rotation of 72° about 0 applied to the reference pentagon and q as a half-turn about OA. pq=p succeeding q. The identity transformation is e. (a) Express a half-turn about OB in terms of p and q. (b) Show that qp=p4q. 9 Transforms and automorphisms In any group it is often found useful to consider the element a- 1ba where a and b are any elements and a- 1 is the inverse of a such that aa- 1=a- 1a=e, the identity element.