How to show two elements in a group are (not) conjugate?

Question

While stuying group theory, I was wondering how to proof two elements in a group are not conjugate?

For example, say we're in the symmetric group $S_8$, and I have to show that the two elements $(1354672)$ and $(18967543)$ are not conjugate. I can see that the orders of both elements are not equal. And order should be conserved under the conjugation map $τ:S_8 → S_8$, since this mapping is an isomorphism. So I think I can conclude that these two elements are not conjugate.

But how do can you prove two elements are not conjugate when they have the same order? Or does same order imply conjugation?

Thanks in advance!

Answer 1

In $S_n$, two elements are conjugate if and only if they have the same cycle type: https://groupprops.subwiki.org/wiki/Cycle_type_of_a_permutation

So, if you have two elements of $S_n$ and you want to know if they are conjugate, you have to determine their cycle type. If they have the same cycle type, they are conjugate, and if they don't have the same cycle type, they are not conjugate.