2024/11/15 by Philippe Tranchida, Tranchida, Philippe
Mathematics · Engineering · #Finite Group Theory Research #Advanced Topics in Algebra #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2411.10299
For q = pn with p an odd prime, the projective linear group PGL(2,q) can be seen as the stabilizer of a conic O in a projective plane π= PG(2,q). In that setting, involutions of PGL(2,q) correspond bijectively to points of π not in O. Triples of involutions \ αP,αQ,αR \ of PGL(2,q) can then be seen also as triples of points \P,Q,R\ of π. We investigate the interplay between algebraic properties of the group H = ⟨ αP,αQ,αR ⟩ generated by three involutions and geometric properties of the triple of points \P,Q,R\. In particular, we show that the coset geometry Γ= Γ(H,(H0,H1,H2)), where H0 = ⟨ αQ,αR ⟩, H1 = ⟨ αP,αR ⟩ and H2 = ⟨ αP,αQ ⟩ is a regular hypertope if and only if \P,Q,R\ is a strongly non self-polar triangle, a terminology we introduce. This entirely characterizes hypertopes of rank 3 with automorphism group a subgroup of PGL(2,q). As a corollary, we obtain the existence of hypertopes of rank 3 with non linear diagrams and with automorphism group PGL(2,q), for any q = pn with p an odd prime. We also study in more details the case where the triangle \P,Q,R\ is tangent to O.