Finite loops arising from projective 3-space

By means of the method of transversals and sections we build up (left) loops (L,*) contained in the group 𝒢 where 𝒢=PGL_2(q) or 𝒢=PSL_2(q) , q>3 odd. Since such groups can be embedded in PG(3,q) , the subgroups we considered for the construction of (L,*) are geometrically relevant (lines or planes of PG(3,q)) or algebraically relevant (maximal subgroups).