Examples of Croups 5 



Instead of operations upon an angle a we can have operations 

 upon the expression cosa+2'sin a where 1, as usual, denotes V — 1. 

 Then s performed upon a in cos a -f- z" sin a gives 



cos (sa) -\-i sm(sa)==- 



cos a-j-z sin a 

 while c upon cos a -j- i sin a gives 



cos (Va)-j-z' sin (ca) = 



COS a-j-z" «>z a' 

 Thus, if r means reciprocal-of, the group 



\ 1, .y, c, 5^, cs, scs, esc, (sc) 2 \ 

 corresponds, operation by operation, to 



1 1) — r> i?', i, — i, — ir, >-. — ij. 



Note that i and r here stand for the operations multiplying- 

 by V — i and taking-reciprocal-of, so that z>= — ri. If we pass 

 to the Argand diagram, by representing x-\~iy by (x, y), then 

 multiplication by i will turn through a right-angle, while r will 

 reflect on the x-axis. We thus reproduce our former geometry. 



For still another representation, let the square belonging to 

 the group have vertices a, b, c, d, in the quadrants I, II, III, IV 

 respectively, and let us agree that {abed) as an operator means 

 to change a, b, c, d into b, c, d, a, advancing cyclically the letters 

 within the symbol. Then the corresponding group on the letters 

 is 



{1, (ab)(cd), (bd), (debd), {abed), (ae), (ad)(bc), (ac)(bd)\. 



The reader will find it interesting to write out the various sub- 

 groups of the above and compare with the corresponding groups of 

 movements of the square. 



Again, we might consider the mutations of the coordinates of 

 a point (x, y) on the terminal line of the angle a. This would 



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