Disjoint cycles in permutations

[RJLiberator]

Homework Statement

Homework Equations

The Attempt at a Solution

My answers for the disjoint cycles of f,g,h,k are as follows. My question concerns my disjoint cycles for g. Are we allowed to have 3 cycles? I would think "yes" because that just makes sense to me.

for f: ( 1 3 8 4 6) (2 5)
g: (2 6 7) (3 8) (4 5)
h: (1 8 7 3 5 2) (4 6)
k: (1 5 4) (2 3 8)

For orders:

order f: lcm(5, 2) = 10
order g: lcm ( 3, 2) = 6
order h: lcm(6, 2) = 6
order k: lcm(3, 3) = 3

 

[RJLiberator]

Also, if I wanted to write f as a product of transpositions I would take:
f = (1 3 8 4 6) (2 5)
product of transpositions: (1,3)(3,8)(8,4)(4,6)(2,5)
5 total so odd permutation