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# Class Note for MATH 409 with Professor Martin at KU

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COURSE
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KARMA
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This 4 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Kansas taught by a professor in Fall. Since its upload, it has received 20 views.

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Date Created: 02/06/15
PERMUTATIONS APPENDIX F Permutations If S is a finite set then a permutation of S is a function fS gtS that has the following two properties 1 if a and b are distinct elements of S then u and fb are also distinct elements of S 2 for every element y of S there is an element x of S such that y fx It is customary to display permutations as a collection of cycles A cycle of a permutation f is a cyclic sequence a1 a2 ak where ll1 ai for i12k 1 and a1fak F1 PERMUTATIONS EXAMPLE Fl If S 1 2 3 4 5 6 7 and f1 6f2 5f3 7f4 4 f5 3 f6 1f7 2 then 1 6 5 3 7 2 and 4 are cycles of f as are 6 1 and 3 7 2 5 However since cycles are by their definition cyclically ordered it follows that 1661 and 5372372572532537 Hence 1 6 5 3 7 2 and 4 are the complete set of cycles of f and we write f 1 65 3 7 24 The order of the cycles is immaterial Thus f1653 7 24 53 7 2416 41653 7 2 3 7 2 546 1 EXAMPLE F2 If S123 4 5 6 7 8 9 a b c and f1af21f3 c f4 8 f5 9 f6 7 f7 3 f8 419 6 fa 2 jib b fc 5 then f 21a73 c 5 9 64 8b If f and g are permutations of the same set S then the composition f g is also a permutation of S such that F2 PERMUTATIONS fgx gx for all x in 5 EXAMPLE F3 If f1 653724 and g17263 54 then gxl g1 f7 2 gxz g2 16 1 16 8X3 g3 f5 3 fg4 g4 f1 6 16 8X5 g5 f4 4 fg6 g6 13 7 fg7 11530 12 5 Consequently fg 1 234 6 7 5 Similarly gf1 NH g6 3 ltgfgt2 gm g5 4 gf3 NS g7 2 gf4 gf4 g4 1 gf5 gf5 g3 5 gf6 gf6 g1 7 g 7 NO g2 6 Consequently gf 1 3 2 456 7 F3 PERMUTATIONS EXERCISES F Rewrite the functions of exercises 1 5 in terms of their cycles f1 62 53 7f4 25 3 61f7 4 f1 62 53 7f4 85 3 61f7 2 f8 4 f1 92 53 7f4 85 3 61f7 2 f8 49 6 f0 910 513 74 816 3 61f7 a 18 49 6 u 2 f0 910 513 bf4 816 3 61f7 a 18 49 6 at 2 2 fb 7 Suppose f 1 2 3 4 5 6 7 8 9 g 4 3 2 159 8 7X6 h 1 23 45 67 l 2 3 8X9 Display the following compositions in terms of their cycles a e i fg b gf c fh d hf gh f hg g ff h gg hh F4

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