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    Orbit-stabilizer theorem

    T-QYDVH

    An orbit has one point for each coset of the stabilizer.Suggest a slogan

    Let a group \(G\) act on a set \(X\), and let \(x\in X\) have orbit \(Gx\) and stabilizer \(G_x\). Then \[ \size{Gx} = [G: G_x], \] and if \(G\) is finite, then \(\size{Gx} = \size{G} / \size{G_x}\).

    The map \(G/G_x\to Gx\), \(gG_x\mapsto gx\), is a well-defined bijection of sets, which gives the equality \(\size{Gx} = [G:G_x]\).

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    • L-DJKXLFixed-point congruence for $p$-groupsuses
    • T-3X5FFSylow's third theorem: numerical constraintsuses
    • T-4CDTTThe class equation and centers of finite $p$-groupsuses

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