Conjugate group elements

Two group elements $x$ and $y$ are conjugate when $y=gx{g}^{-1}$ for some group element $g$, equivalently when they belong to the same conjugacy class. Conjugate elements have the same order of a group element.

Two permutations in the same symmetric group are conjugate if $\tau =\pi \sigma {\pi }^{-1}$ for some permutation $\pi$. Conjugation relabels every permutation cycle, so it preserves cycle type. Conversely, equal cycle types permit a relabelling between corresponding cycles, establishing conjugacy.