The stabilizer of any four-point subset of the projective line is isomorphic to V4, D4, A4, or S4, with the specific case determined by the field characteristic and a polynomial splitting condition.
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On Groups of Linear Fractional Transformations Stabilizing Finite Sets of Four Elements
The stabilizer of any four-point subset of the projective line is isomorphic to V4, D4, A4, or S4, with the specific case determined by the field characteristic and a polynomial splitting condition.