2. Consider the symmetric group S7. Let f_ 1 2 3 4 5 6 7 _ 1 2 _ 3 5 4 1 2 6 7 ' 9 3 7 (a) Write f and g as the product of disjoint cycles. 34 6 54 1...

2. Consider the symmetric group S7. Let

f=  , g = 

(a) Write f and g as the product of disjoint cycles.

(b) Find the order of f and g. 

(c) Determine the parity of f and g.

(d) Find fg-1.

(e) Does S7 have an element of order 10? If so, give an example of such an element, if not, explain why.

(f) Does S7 have an element of order 11? If so, give an example of such an element, if not, explain why. 

2. Consider the symmetric group S7. Letf_ 1 2 3 4 5 6 7 _ 1 2_ 3 5 4 1 2 6 7 ’ 9— 3 7 (a) Write f and g as the product of disjoint cycles. 34 654 1 3) 56 (b) Find the order of f and g.(c) Determine the parity of f and g.(d) Find fg-l. (e) Does 87 have an element of order 10? If so, give an example of such anelement, if not, explain why. (f) Does 87 have an element of order 11? If so, give an example of such anelement, if not, explain Why.