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 $(x_{1},\ldots,x_{p})$ for which $x_{1}\cdots x_{p}=1$. 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 $(x_{1},\ldots,x_{p})$ to $(x_{m+1},\ldots,x_{p},x_{1},\ldots,x_{m})$. By the Orbit-Stabilizer Theorem, each orbit contains exactly $1$ or $p$ tuples. Since $(1,\ldots,1)$ 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 $(x_{1},\ldots,x_{p})$ which is left fixed by every element of $\mathbb{Z}/p\mathbb{Z}$. For this tuple we have $x_{1}=\ldots=x_{p}$, and so $x_{1}^{p}=x_{1}\cdots x_{p}=1$, and $x_{1}$ is therefore an element of order $p$.

References

• 1 James H. McKay. Another Proof of Cauchy’s Group Theorem, American Math. Monthly, 66 (1959), p119.
Title proof of Cauchy’s theorem ProofOfCauchysTheorem 2013-03-22 12:23:30 2013-03-22 12:23:30 yark (2760) yark (2760) 8 yark (2760) Proof msc 20E07 msc 20D99