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Math Help - Fixed point theorem p-groups

  1. #1
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    Fixed point theorem p-groups

    I have no idea where to even start.

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  2. #2
    MHF Contributor Swlabr's Avatar
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    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.
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  3. #3
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    If G acts on a finite set S, then S can be written as a disjoint union (See Hungerford's algebra p 93)
    S=S_0 \cup \bar{x_1} \cup \bar{x_2} \cup \cdots \cup \bar{x_n}, where S_0=\{x \in S: g\cdot x=x \text{ for all } g \in G\}, |\bar{x_i}|=|G:G_{x_i}|, and |\bar{x_i}|>1 for all i. We see that |\bar{x_i}| 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 S_0 is not empty and p \nmid|S_0|. An orbit of x \in S, denoted \bar{x}, has a single element iff x \in S_0. Can you conclude it from here?
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