Conjugation preserves order

Proposition (Conjugation preserves order). Let $G$ be a group . For $x\in G$, the order of $x$, denoted $\mathrm{ord}(x)$, is the least positive integer $n$ such that $x^n=e$ (if such an $n$ exists), and $\mathrm{ord}(x)=\infty$ otherwise. If $x,y\in G$ are conjugate , i.e. $y=gxg^{-1}$ for some $g\in G$, then $\mathrm{ord}(y)=\mathrm{ord}(x)$.

Context. Many group-theoretic invariants are constant on conjugacy classes. Order is the first basic example and is used, for instance, in the class equation and Sylow theory.

Proof sketch. For every integer $n\ge 1$ one has

Hence $(gxg^{-1})^n=e$ iff $x^n=e$, so the minimal such $n$ (or lack thereof) agrees for $x$ and its conjugate.