Let A = {1, 2, 3, 4, 5, 6}. Recall that a bijection p: A rightarrow A is called

Question

Let A = {1, 2, 3, 4, 5, 6}. Recall that a bijection p: A rightarrow A is called a permutation of A. a) Give an example of a permutation p of A such that p(a) a for all a elementof A. Represent your permutation p as a set of ordered-pairs. b) Express the 4-fold composition p^4 = p p p p: A rightarrow A of your permutation p of (a) as a set of ordered-pairs. c) Express the inverse of your permutation p of (a) as a set of ordered-pairs. d) What is the smallest positive integer n such that the the n-fold composition p^n = p p ..... p: A rightarrow A of your permutation p of (a) is the identity function 1_A?

Explanation / Answer

a) p = { (1,2), (2,3), (3, 4), (4, 5), (5, 6), (6, 1) }

b) p*p*p*p = {(1,5),(2,6),(3,1),(4,2),(5,3),(6,4)}

c) inv(p) = { (1, 6), (2, 1), (3, 2), (4, 3), (5, 4), (6, 5)}

d) Smallest period = 6. Because p is a cycle of size 6, every element is moved in p and every element can map to any other element by successive application of p.

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