Find the composition of the following cycles representing pe

Find the composition of the following cycles representing permutations on A = {1, 2, 3, 4, 5}. Write your answer as a composition of one or more disjoint cycles.

a. (2, 4, 5, 3) (1, 3)

b. (3, 5, 2) (2, 1, 3) (4, 1)

c. (2, 4) (1, 2, 5) (2, 3, 1) (5, 2)

Solution

Definition: A permutation of a set X is a rearrangement of its elements.

Let X = {1, 2, 3, 4}. Then there are 24 permutations: 1234, 1243, 1324, 1342, 1423, 1432

2134, 2143, 2314, 2341, 2413, 2431

3214, 3241, 3124, 3142, 3421, 3412

4231, 4213, 4321, 4312, 4123, 4132

Definition: Let X = {1, 2, . . . , n} and : X X be a permutation. Let i1, i2, . . . , ir

be distinct numbers from {1, 2, . . . , n}. If (i1) = i2, (i2) = i3, . . . , (ir1) = ir, (ir) = i1, and (i) = i for other numbers from {1, 2, . . . , n},

then is called an r-cycle.

An r-cycle is denoted by (i1 i2 . . . ir).

1 1 = (1) 1 cycle

1 2 1 2 = (1) 1 cycle 1 2 2 1 = (12) 2 cycle 1 2 3 3 2 1 = (13) 2 cycle 1 2 3 2 3 1 = (123) 3 cycle 1 2 3 4 4 3 1 2 = (1423) 4 cycle 1 2 3 4 5 3 5 4 2 1 = (13425) 5 cycle 1 2 3 4 5 2 5 3 4 1 = (125) 3 cycle 1 2 3 4 5 2 5 4 3 1 is not a cycle

Let = 1 2 3 2 3 1 , = 1 2 3 2 1 3 . We have: = 1 2 3 2 3 1 1 2 3 2 1 3 = 1 2 3 3 2 1 , = 1 2 3 2 1 3 1 2 3 2 3 1 = 1 2 3 1 3 2