Finite subgroups of $\operatorname{PGL}_2(K)$ arising from configurations of skew lines in $\mathbb{P}^3_K$

Abstract: We study finite groups arising from configurations of skew lines in $\mathbb{P}<sup>3_K$. Given a finite set ${L}$ of pairwise skew lines in $\mathbb{P}<sup>3_K$ and the associated groupoid $C_{L}$, we consider the endomorphism group $$G_{L} \subset \operatorname{Aut}(L_i) \cong \operatorname{PGL}<em>2(K)$$ for any line $L_i \in {L}$, and we ask which finite subgroups of $\operatorname{PGL}_2(K)$ can occur in this way. Using a matrix description of skew lines, we express the generators of $G</em>{L}$ in terms of a family of matrices $M_i \in \operatorname{GL}<em>2(K)$ and analyze $G</em>{L}$ in the abelian and non-abelian cases. In the abelian situation we show that, after a change of basis, the matrices $M_i$ are simultaneously upper triangular and we obtain explicit families realizing cyclic groups and $p$-semi-elementary groups of the form $C_p^m \rtimes C_n$. In the non-abelian case we prove that no dihedral group $D_n$ with $n \ge 3$ can occur, while we construct configurations with $G_{L} \cong A_4, S_4, A_5$ and describe their orbit structure. Viewed through the lens of $(a,b)$-geproci sets, these results provide a group-theoretic description of collinearly complete point sets and yield new examples of half-grid geproci sets.