Finite loops arising from projective 3-space (Q1702156)

In this paper, the authors apply the technique introduced by \textit{R. Baer} [Trans. Am. Math. Soc. 46, 110--141 (1939; Zbl 0022.01105)] ``where he shows that the theory of nets is completely equivalent to a well-determined chapter in the theory of groups. To do this he investigates certain groups of net transformation'', to construct left loops, loops an groups as transversals of the left coset space \(G/D\), for suitable subgroups \(D\) in \(G=\mathrm{PGL}_2(q)\) or \(\mathrm{PSL}_2(q)\), \(q=ph>3\) odd. The aim of the authors is to get such algebraic structures starting from transversals having particular geometrical meaning inside the kinematic space \(\mathrm{PGL}_2(q)\) or \(\mathrm{PSL}_2(q)\) embedded in the projective space \(\mathrm{PG}(3,q)\). In all cases considered by the authors, the subgroup \(D\) is either a line or a maximal subgroup of \(\mathrm{PSL}_2(q)\) and the transversal \(L=H\cup Ha\) is the union of either a maximal subgroup and a left coset of it or two disjoint lines.