# S12 Conjugate Elements

• Feb 22nd 2009, 03:23 PM
d_p_osters
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.
• Feb 22nd 2009, 06:36 PM
ThePerfectHacker
Quote:

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 ${{12}\choose 5}$ ways to choose elements for a 5-cycle. Once we have chosen these 5 elements there are $4!$ 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, ${7\choose 2}$ and once those have been chosen we have ${5\choose 2}$ for the remaining 2-cycle. Therefore, we get $\frac{1}{2}\cdot 4! {{12}\choose 5}{7\choose 2}{5\choose 2}$ because we overcounted the 2-cycles twice i.e. $(12345)(67)(89)$ is the same as $(12345)(89)(67)$.