This is for n greater than or equal to 4 and we are working in Sn.
For example with n=4, we cannot write them as a product of 3-cycles, but we can write them as a product of 2-cycles. Thus, transpositions cannot be written as a product of 3-cycles.
I am unsure of how to “show” this. Any help or suggestions would be greatly appreciated.
A 3 cycle can be broken down to 2 transpositions. Any combination of an even number of transpositions is an even number of transpositions.
– Doug M
2 days ago
Transpositions are odd permutations, and 33-cycles are even.