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

TL;DR

This paper classifies the groups of linear fractional transformations that stabilize a set of four elements on the projective line over various fields, revealing their isomorphism types depend on the field's characteristic.

Contribution

It provides a complete classification of the stabilizer groups for four-element sets, extending known results from smaller sets to this specific case.

Findings

Stabilizer groups are isomorphic to V_4, D_4, A_4, or S_4 depending on the field characteristic.

The classification depends on the algebraic properties of the underlying field.

The work bridges group theory and projective geometry in the context of finite set stabilization.

Abstract

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.