I have no idea where to even start.

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- February 23rd 2011, 10:26 PMCropDusterFixed point theorem p-groups
I have no idea where to even start.

http://img580.imageshack.us/img580/6...10224at125.png - February 24th 2011, 12:41 AMSwlabr
Hint: By Burnside's Lemma (aka, the Lemma Which Is Not Burnside's), the order of G must divide the sum of the orders of the orbits.

- February 24th 2011, 01:31 AMTheArtofSymmetry
If G acts on a finite set S, then S can be written as a disjoint union (See Hungerford's algebra p 93)

, where , , and for all i. We see that is divisible by p by Lagrange's theorem, since a p-group G acts on S. By assumption, S is not divisible by p, so is not empty and . An orbit of , denoted , has a single element iff . Can you conclude it from here?