# Show that a transposition cannot be written as a product of 3-cycles.

Show that, working in $Sn$ with $n \geq 4$, a transposition cannot be written as a product of 3-cycles.

With $n=4$, for instance, 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 not sure how to prove 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 Oct 22 '16 at 0:29
• Transpositions are odd permutations, and $3$-cycles are even. – user26857 Oct 22 '16 at 7:00