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?

Try an $n$-cycle if $n>2$. – Jyrki Lahtonen Feb 15 '20 at 20:08 — Jyrki Lahtonen, Feb 15 '20 at 20:08

score 2 · Accepted Answer · answered Feb 15 '20 at 20:10

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.

Does every permutation commute with some transposition?

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

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Does every permutation commute with some transposition?