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APPENDIX F
Permutations
If S is a finite set, then a permutation of S is a function f:S → S that has the following two properties:
- if a and b are distinct elements of S then f(a) and f(b) are also distinct elements of S;
- for every element y of S there is an element x of S such that y = f(x). It is customary to display permutations as a collection of cycles. A cycle of a permutation f is a cyclic sequence ( a 1 a 2 … ak ) where ai +1 = f(ai) for i = 1, 2, …, k - 1 and a 1 = f(ak).
EXAMPLE F.1 If S = {1, 2, 3, 4, 5, 6, 7} and f (1) = 6, f (2) = 5, f (3) = 7, f (4) = 4,
f (5) = 3, f (6) = 1, f (7) = 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 (1 6) = (6 1) and (5 3 7 2) = ( 3 7 2 5) = (7 2 5 3) = ( 2 5 3 7). Hence (1 6), (5 3 7 2), and (4) are the complete set of cycles of f and we write f = (1 6)(5 3 7 2)(4). The order of the cycles is immaterial. Thus, f = (1 6)(5 3 7 2)(4) = (5 3 7 2)(4)(1 6) = (4)(1 6)(5 3 7 2) = (3 7 2 5)(4)(6 1) = ….
EXAMPLE F.2 If S = {1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c } and f (1) = a , f (2) = 1, f (3) =
c , f (4) = 8, f (5) = 9, f (6) = 7, f (7) = 3, f (8) = 4, f (9) = 6, f ( a ) = 2, f ( b ) = b , f ( c ) = 5, then f = (2 1 a )(7 3 c 5 9 6)(4 8)( b ). If f and g are permutations of the same set S , then the composition f+g is also a permutation of S such that
EXERCISES F
Rewrite the functions of exercises 1-5 in terms of their cycles.
- f (1) = 6, f (2) = 5, f (3) = 7, f (4) = 2, f (5) = 3, f (6) = 1, f (7) = 4.
- f (1) = 6, f (2) = 5, f (3) = 7, f (4) = 8, f (5) = 3, f (6) = 1, f (7) = 2, f (8) = 4.
- f (1) = 9, f (2) = 5, f (3) = 7, f (4) = 8, f (5) = 3, f (6) = 1, f (7) = 2, f (8) = 4, f (9) = 6.
- f (1) = 9, f (2) = 5, f (3) = 7, f (4) = 8, f (5) = 3, f (6) = 1, f (7) = a , f (8) = 4, f (9) = 6, f ( a ) = 2.
- f (1) = 9, f (2) = 5, f (3) = b , f (4) = 8, f (5) = 3, f (6) = 1, f (7) = a , f (8) = 4, f (9) = 6, f ( a ) = 2, f ( b ) = 7.
- Suppose f = (1 2 3 4 5 6 7 8 9), g = (4 3 2 1)(5)(9 8 7)(6), h = (1 2)(3 4)(5 6)( 8)(9). Display the following compositions in terms of their cycles. a) f+g b) g+f c) f+h d) h+f e) g+h f) h+g g) f+f h) g+g i) h+h
