Recall that element a of a group G with identity element e has order r > 0 if a^r = e and no smaller positive power of a is the identity. Consider the group S₈. 

a. What is the order of the cycle ( 1, 4, 5, 7)? 

b. State a theorem suggested by part (a). 

c. What is the order of a = (4, 5)(2, 3, 7)? of r = (1, 4)(3, 5, 7, 8)? 

d. Find the order of each of the permutations given in Exercises 10 through 12 by looking at its decomposition into a product of disjoint cycles. 

e. State a theorem suggested by parts (c) and (d).

Data from Exercise 10 

 Data from Exercise 11

Data from Exercise 12

Transcribed Image Text:

1 8 4 5 6 7 8 4 5 1 2 3 2637