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On Wikipedia, I have seen a nice proof of Zagier for the existence of representations of primes as sum of two squares:

http://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squares#Zagier.27s_.22one-sentence_proof.22

The proof uses the (trivial) fact that the number of fixed points of an involution of a finite set has the same parity as the cardinality of the set itself.

Do you know any other examples for existence proofs using the parity of fixed points of involutions?

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A more general fact is that the number of fixed points of the action of a finite $p$-group on a finite set has the same cardinality $\bmod p$ as the set itself. This is used, for example, to prove that finite $p$-groups have nontrivial center and it is also part of the proof of the Sylow theorems.

While this isn't an existence proof, it can also be used to give a short proof of Fermat's little theorem: given a positive integer $a$, consider the action of $\mathbb{Z}/p\mathbb{Z}$ by cyclic permutation on the set of words of length $p$ on an alphabet of size $a$. There are $a$ fixed points and $a^p$ words, hence $a^p \equiv a \bmod p$.

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    @myself: that is the famous McKay proof of Cauchy's theorem. I like it too...2011-05-19