Question

Please help on groups and symmetries

1a) Find three elements in S7 or order 6 such that each has a different cycle type?

b) How many elements in S9 have order 10?

Answer 1

By order 6, do you mean something like
(1,2)(3,4,5) ?

Answer 2

for a, I'd say
(1,2)(3,4,5)
(1,2,3,4,5,6)
(2,3,4,5,6,7)(1,5)

Answer 3

for b, I'm not too sure about this, but as there are no 10-cycles in S9, they'd have to be the products of disjoint 5- and 2- cycles.

There are 9p5 5-cycles times 4p2 or 12 2-cycles (transpositions), so their product would be it