Does *every* permutation commute with *some* transposition?

Question

I'm getting started with the study of permutations on $n$ objects, and I'm trying to prove/disprove something that is probably trivial: does every permutation commute with some transposition?

Answer 1

No. If you are permuting $n$ objects a cycle of length $n$ will not commute with any transposition - except when $n=2$ when the only cycle and the only transposition are the same element. ($n=0, 1$ have no transpositions so fail for a trivial reason)

I will leave you to explore the details.