Group theory: Conjugates, commutativity

[Kate2010]

Homework Statement

In S8, consider p= (1 3)(2 4 5 6) and q=(1 3)(2 4)(7 6 5)

1. Find the number of conjugates of p and the number of conjugates of q
2. Find the number of permutations that commute with p and the number that commute with q

Homework Equations

The Attempt at a Solution

I think I've got myself really confused.

1. For p, we are working in S8 so I initally count 8!
I then divide by 2 as (1 3) = (3 1) and similarly divide by 4 because of the 4-cycle.
The disjoint cycles are not the same length so my final answer is 8! / (4 x 2) = 7!

For q, we have 8! / (2 x 2 x 3 x2) = 1680 where the final 2 is because we have 2 2-cycles.

2. I'm not so sure about how to do this. I think it is asking me to find the size of the centralizer of each element but I don't know how to do this.

 

[jamiem10]

I think you need to divide by 2 again for p because 7 and 8 could either be mapped to themselves or to each other without it affecting anything else