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

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.

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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.
– user26857
yesterday

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