We study finite groups arising from configurations of skew lines in $\mathbb{ P}^3_K$. Given a finite set ${ L}$ of pairwise skew lines in $\mathbb{ P}^3_K$ and the associated groupoid $C_{ L}$, we consider the endomorphism group $G_{ L} \subset \operatorname{ Aut}(L_i) \cong \operatorname{ PGL}_2(K)$ 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 $G_{ L}$ in terms of a family of matrices $M_i \in \operatorname{ GL}_2(K)$ and analyze $G_{ L}$ 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.
Categories: math.AG math.AC math.GR