On Groups of Linear Fractional Transformations Stabilizing Finite Sets of Four Elements

Let $E$ be a subset of the projective line over a commutative field $\mathbb{K}$. When $\mathbb{K}$ has infinite cardinality, it is well known that if $E$ contains at most three elements, then the group of linear fractional transformations preserving $E$ is either infinite or isomorphic to the symmetric group on three elements. In this work, we investigate the case where $E$ consists of four elements. We show that the group of projective linear transformations stabilizing $E$ is, depending on the characteristic of the field $\mathbb{K}$, isomorphic to either the Klein four-group $V_4$, the dihedral group $D_4$ of order eight, the alternating group $\mathfrak{A}_4$ of order twelve, or the symmetric group $\mathfrak{S}_4$ of order twenty-four.