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 of the cyclic group $\mathbb{Z}/p\mathbb{Z}$ on $X$ under which $m\in \mathbb{Z}/p\mathbb{Z}$ sends the 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 $p$.

Title	proof of Cauchy’s theorem
Canonical name	ProofOfCauchysTheorem
Date of creation	2013-03-22 12:23:30
Last modified on	2013-03-22 12:23:30
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	8
Author	yark (2760)
Entry type	Proof
Classification	msc 20E07
Classification	msc 20D99