1. S12 Conjugate Elements

I'm trying to find the number of elements conjugate to (12)(34)(56789) in S12 but don't know how to go about it.

Any help is much appreciated.

2. Originally Posted by d_p_osters
I'm trying to find the number of elements conjugate to (12)(34)(56789) in S12 but don't know how to go about it.

Any help is much appreciated.
First we form a 5-cycle, there are 12 elements and we only choose 5 of them therefore there are $\displaystyle {{12}\choose 5}$ ways to choose elements for a 5-cycle. Once we have chosen these 5 elements there are $\displaystyle 4!$ ways of ordering them to form distinct 5-cycles. Now we need to form a 2-cycle, there are 7 elements left and we choose 2, therefore, $\displaystyle {7\choose 2}$ and once those have been chosen we have $\displaystyle {5\choose 2}$ for the remaining 2-cycle. Therefore, we get $\displaystyle \frac{1}{2}\cdot 4! {{12}\choose 5}{7\choose 2}{5\choose 2}$ because we overcounted the 2-cycles twice i.e. $\displaystyle (12345)(67)(89)$ is the same as $\displaystyle (12345)(89)(67)$.