Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 365–388 On the rank two geometries of the groups PSL(2; q): part II∗ Francis Buekenhout , Julie De Saedeleer , Dimitri Leemans y Universite´ Libre de Bruxelles, Departement´ de Mathematiques´ - C.P.216 Boulevard du Triomphe, B-1050 Bruxelles Received 30 November 2010, accepted 25 January 2013, published online 15 March 2013 Abstract We determine all firm and residually connected rank 2 geometries on which PSL(2; q) acts flag-transitively, residually weakly primitively and locally two-transitively, in which 0 one of the maximal parabolic subgroups is isomorphic to A4, S4, A5, PSL(2; q ) or PGL(2; q0), where q0 divides q, for some prime-power q. Keywords: Projective special linear groups, coset geometries, locally s-arc-transitive graphs. Math. Subj. Class.: 51E24, 05C25 1 Introduction In [5], we started the classification of the residually weakly primitive and locally two- transitive coset geometries of rank two for the groups PSL(2; q). The aim of this paper is to finish this classification. It remains to focus on the cases in which one of the max- 0 0 imal parabolic subgroups is isomorphic to A4, S4, A5, PSL(2; q ) or PGL(2; q ) where q0 divides q. For motivation, basic definitions, notations and context of the work we refer to [5]. In Section 3, we sketch the proof of our main result: ∼ Theorem 1.1. Let G = PSL(2; q) and Γ(G; fG0;G1;G0\G1g) be a locally two-transitive 0 RWPRI coset geometry of rank two. If G0 is isomorphic to one of A4, S4, A5, PSL(2; q ) or PGL(2; q0), where q0 divides q, then Γ is isomorphic to one of the geometries appearing in Table1, Table2, Table3, Table4, Table5, and Table6. ∗This paper is a part of SIGMAP’10 special issue Ars Math. Contemp. vol. 5, no. 2. ySupported by the “Communaute´ Franc¸aise de Belgique - Actions de Recherche Concertees”´ E-mail addresses: [email protected] (Francis Buekenhout), [email protected] (Julie De Saedeleer), [email protected] (Dimitri Leemans) Copyright c 2013 DMFA Slovenije 366 Ars Math. Contemp. 6 (2013) 365–388 r G0 =∼ A5 q = 4 with r prime G0 \ G1 G1 ] Geom. ] Geom. Extra conditions loc. (G; s)- up to conj. up to isom. on q arc-trans. g. q±1 Γ1 D10 D30 1 1 15 odd s = 3 Γ2 A4 E16 :3 1 1 q = 16 s = 3 Γ3 A4 E16 :3 5 2 q = 64 s = 3 4r−1−1 2(4r−2−1)+3:2r−2 Γ4 A4 E16 :3 3 3r r > 3, r odd prime s = 3 G0 =∼ A5 q = p = ±1(5) with p odd prime G0 \ G1 G1 ] Geom. ] Geom. Extra conditions loc. (G; s)- up to conj. up to isom. on q arc-trans. g. Γ5 D10 D20 2 1 q = ±1(20) s = 3 Γ6 D10 D30 2 1 q = ±1(30) s = 3 q±1 Γ7 D10 A5 2 1 10 even s = 2 q±1 Γ8 D10 A5 1 1 10 odd s = 2 Γ9 A4 S4 2 1 q = ±1(40) or q = ±9(40) s = 3 Γ10 A4 A5 2 1 q = ±1(40) or q = ±9(40) s = 2 Γ11 A4 A5 1 1 q = ±11(40) or q = ±19(40) s = 2 2 G0 =∼ A5 q = p = −1(5) with p odd prime G0 \ G1 G1 ] Geom. ] Geom. Extra conditions loc. (G; s)- up to conj. up to isom. on q arc-trans. g. Γ12 D10 D20 2 1 q = −1(20) s = 3 Γ13 D10 D30 2 1 q = −1(30) s = 3 q+1 Γ14 D10 A5 2 1 10 even s = 2 q+1 Γ15 D10 A5 1 1 10 odd s = 2 Γ16 A4 S4 2 1 q = −1(40) or q = 9(40) s = 3 Γ17 A4 A5 2 1 q = −1(40) or q = 9(40) s = 2 Γ18 A4 A5 1 1 q = −11(40) or q = 19(40) s = 2 ∼ Table 1: The RWPRI and (2T )1 geometries with G0 = A5. G0 =∼ A4 q = p > 3 and q = 3; 13; 27; 37(40) or q = 5 G0 \ G1 G1 ] Geom. ] Geom. Extra conditions locally(G; s)-arc- up to conj. up to isom. on q transitive graphs Γ1 3 Z6 1 1 q = 13; 37; 83; 107(120) s = 3 q+1 q+1 6 +1 q+1 Γ2 3 D6 6 2 6 odd s = 3 q−1 q−1 6 +1 q−1 Γ3 3 D6 6 2 6 odd s = 3 q+1 q+1 q+1 Γ4 3 D6 6 12 6 even s = 3 q−1 q−1 q−1 Γ5 3 D6 6 12 6 even s = 3 q+1 q+1 Γ6 3 A4 3 − 1 6 3 j q + 1 s = 2 q−1 q−1 Γ7 3 A4 3 − 1 6 3 j q − 1 s = 2 ∼ Table 2: The RWPRI and (2T )1 geometries with G0 = A4 F. Buekenhout et. al.: On the rank two geometries of the groups PSL(2; q): part II 367 G0 =∼ S4 q = p > 2 and q = ±1(8) G0 \ G1 G1 ] Geom. ] Geom. Extra conditions locally(G; s)-arc- up to conj. up to isom. on q transitive graphs Γ1 D6 D12 2 1 q = ±1(24) s = 3 Γ2 D6 D18 2 1 q = ±1(72) or q = ±17(72) s = 3 q±1 Γ3 D6 S4 2 1 6 even s = 2 q±1 Γ4 D6 S4 1 1 6 odd s = 2 Γ5 D8 D16 2 1 q = ±1(16) s = 7 Γ6 D8 D24 2 1 q = ±1(24) s = 3 q±1 Γ7 D8 S4 2 1 8 even s = 4 q±1 Γ8 D8 S4 1 1 8 odd s = 4 Γ9 A4 A5 2 1 q = ±1(40) or q = ±9(40) s = 3 ∼ Table 3: The RWPRI and (2T )1 geometries with G0 = S4. n nm G0 =∼ PSL(2; 2 ) q = 2 , with m prime G0 \ G1 G1 ] Geom. ] Geom. Extra conditions loc.(G; s)-arc- up to up to on q trans. conj. isom. graphs n n Γ1 E2n :(2 − 1) E2mn :(2 − 1) 1 1 m = 2, n 6= 1 s = 3 Γ2 2 D6 1 1 q = 4; n = 1, m = 2 s = 2 2 Γ3 2 2 1 1 q = 4; n = 1, m = 2 s = 3 Γ4 3 A4 1 1 q = 4; n = 1, m = 2 s = 3 m q±1 Γ5 D10 D30 1 1 q = 4 ; n = 2; 15 odd s = 3 ∼ n Table 4: The RWPRI and (2T )1 geometries with G0 = PSL(2; 2 ). nm G0 =∼ q = p , p and PSL(2; pn) m odd primes G0 \ G1 G1 ] Geom. ] Geom. Extra conditions locally(G; s)-arc- up to conj. up to isom. on q transitive graphs m−1 3m−1−1 m Γ1 3 A4 3 − 1 2m q = 3 ; n = 1, m 6= 3 s = 2 Γ2 3 A4 8 2 q = 27; n = 1, m = 3 s = 2 ∼ 0 0 Table 5: The RWPRI and (2T )1 geometries with G0 = PSL(2; q ), q odd. n 2n G0 =∼ PGL(2; p ) q = p , with p odd prime G0 \ G1 G1 ] Geom. ] Geom. Extra conditions loc. (G; s)-arc- up to conj. up to isom. on q transitive graphs n n Γ1 Epn :(p − 1) Ep2n :(p − 1) 2 1 none s = 3 n Γ2 PSL(2; p ) A5 2 1 q = 9 s = 3 Γ3 D8 PGL(2; 3) 1 1 q = 9 s = 4 ∼ 0 Table 6: The RWPRI and (2T )1 geometries with G0 = PGL(2; q ). 368 Ars Math. Contemp. 6 (2013) 365–388 Observe that, geometry Γ5 in Table4 is exactly geometry Γ1 in Table1. In Section 4, we recall the subgroup lattice of PSL(2; q), and we give the two-transitive representations of the maximal subgroups. In Section 5, we prove Theorem 1.1, which is based on the proof of Propositions 5.5, 5.6, 5.10, 5.12, 5.16 and 5.21. For that purpose, we determine the rank two RWPRI and (2T )1 geometries of PSL(2; q) and their number, up to isomorphism and up to conjugacy. The existence of such geometries is equivalent to the existence of a locally 2-arc transitive bipartite graph for which the action of G is primitive on one of the bipartite halves (see [8]). Our result is also a part of the program initiated in [8]. These graphs are interesting in their own right because of the numerous connections they have with other fields of mathematics (see [8] for more details). We also refer to the classification of these graphs for almost simple groups with socle a Ree simple group Ree(q) (see [7]). In terms of locally 2-arc-transitive graphs, we obtain here the classifica- tion of these graphs with one vertex-stabilizer maximal in PSL(2; q) and isomorphic to A4, 0 0 S4, A5, PSL(2; q ) or PGL(2; q ). The last column of Table1, Table2, Table3, Table4, Table5 and Table6 gives, for each geometry Γ, the value of s such that Γ is a locally s-arc-transitive but not a locally (s + 1)-arc-transitive graph. In section 6, we determine the exact value of s in all cases that are not current by the method of Leemans.