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 ways to choose elements for a 5-cycle. Once we have chosen these 5 elements there are 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, and once those have been chosen we have for the remaining 2-cycle. Therefore, we get because we overcounted the 2-cycles twice i.e. is the same as .