proof of Cauchy’s theorem

Let $G$ be a finite group, and suppose $p$ is a prime divisor of $|G|$.Consider the set $X$ of all $p$-tuples $(x_1,\mathrm{\dots },x_p)$for which $x_1\mathrm{\cdots }{x}_p=1$.Note that $|X|={|G|}^{p-1}$ is a multiple of $p$.There is a natural group action ofthe cyclic group $\mathbb{Z}/p\mathbb{Z}$ on $X$under which $m\in \mathbb{Z}/p\mathbb{Z}$ sendsthe tuple $(x_1,\mathrm{\dots },x_p)$to $(x_{m+1},\mathrm{\dots },x_p,x_1,\mathrm{\dots },x_m)$.By the Orbit-Stabilizer Theorem, each orbit contains exactly $1$ or $p$ tuples.Since $(1,\mathrm{\dots },1)$ has an orbit of cardinality $1$,and the orbits partition $X$,the cardinality of which is divisible by $p$,there must exist at least one other tuple $(x_1,\mathrm{\dots },x_p)$which is left fixed by every element of $\mathbb{Z}/p\mathbb{Z}$.For this tuple we have $x_1=\mathrm{\dots }=x_p$,and so $x_1^p=x_1\mathrm{\cdots }{x}_p=1$,and $x_1$ is therefore an element of order $p$.