Cauchy’s theorem

This is a really classical result, but this proof is just so magical that I have to include it.

Lemma. Let G be a finite group, and let p be a prime dividing \# G. Then G has an element of order p.

Proof. Consider the action of \Z/p\Z = \langle \sigma \rangle on

    \[ X = \{(g_1, \ldots, g_p) \in G^p\ |\ g_1 \cdots g_p = 1\} \]

given by

    \[ \sigma \cdot (g_1, \ldots, g_p) = (g_2, \ldots, g_p, g_1). \]

Then \# (X^{\Z/p\Z}) \equiv \# X \pmod{p}, since the orbits of size > 1 all have size p (and the fixed points X^{\Z/p\Z} are exactly the union of the orbits of size 1). The right hand side is 0 \pmod{p} as p \mid \# G. Thus,

    \[ \# (X^{\Z/p\Z}) \equiv 0 \pmod{p}. \]

But there is a bijection

    \begin{align*} \{g \in G\ |\ g^p = 1\} &\rA X^{\Z/p\Z}\\ g &\rM (g, \ldots, g). \end{align*}

The former trivially contains the element 1, hence (since its size is divisible by p) it has to contain some other element g. This element will then have (exact) order p. \qedsymbol

Remark. Of course, the result also follows from the much stronger Sylow theorems (of which we only need the existence statement).

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