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    A $p$-group acting on a set of cardinality not divisible by $p$ has a fixed point

    P-WKGRZ

    Area
    Algebra
    Topics
    Group Actions, Orbit-Stabilizer, p-Groups
    Status
    draft

    Let a finite \(p\)-group \(G\) act on a finite set \(X\) with \(p\nmid |X|\). Prove that the action has a fixed point.

    Solution

    Decompose \(X\) into \(G\)-orbits. By orbit–stabilizer, every orbit has cardinality \[ |G:G_x|, \] which is a power of \(p\).

    An orbit has size \(1\) exactly when its point is fixed by all of \(G\). Every nontrivial orbit therefore has cardinality divisible by \(p\).

    Let \(X^G\) denote the set of fixed points. Summing orbit sizes gives \[ |X|\equiv |X^G|\pmod p. \] Since \(p\nmid |X|\), we have \[ |X^G|\not\equiv0\pmod p. \] In particular \(X^G\ne\varnothing\), so the action has a fixed point.

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