How do two conjugate elements of a group have the same order? I'm reading group action in textbook Algebra by Saunders MacLane and Garrett Birkhoff.



I have a problem of understanding the last sentence:

Since conjugation is an automorphism, any two conjugate elements have the same order.

Assume $x,y \in G$ are conjugate, then they are equivalent. As such, $gxg^{-1} = y$ for some $g \in G$. This means $gx = yg$. From here, I could not get how $x,y$ have the same order.
Could you please elaborate on this point?

Since $y=gxg^{-1}$, we have
$$\begin{align}
y^n&=\underbrace{(gxg^{-1})\dots(gxg^{-1})}_{n\text{ times}}\\
&=\underbrace{g\cdot x\cdot (g^{-1}g)\cdot\dots\cdot (g^{-1}g)\cdot x \cdot g^{-1}}_{n\text{ times }x}\\
&=gx^ng^{-1},
\end{align}$$
so, if $x^n=e$, then $y^n=e$, and vice versa (by the inverse of conjugation).
